# Copyright 2026 Philippe Billet assisted by LLMs in free mode: chatGPT, Qwen, Deepseek, Gemini, Claude, le chat Mistral.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
hamiltonian_catalog.py — Hamiltonian catalog for analysis, visualization and more...
=========================================================================================
Extended catalog of 1D and 2D Hamiltonians for pseudodifferential
and semiclassical analysis, visualization, and symbolic exploration.
Each entry:
key : {
"expr" : sympy.Expr,
"dim" : 1 or 2,
"category" : str,
"description" : str
}
Usage:
from hamiltonian_catalog_extended import get_hamiltonian
H, vars, meta = get_hamiltonian("henon_heiles")
"""
import sympy as sp
import numpy as np
from collections import Counter
import itertools
import os
import json
# ---------------------------------------------------------------------
# SYMBOLS (shared)
# ---------------------------------------------------------------------
# Real variables (real=True)
x, y, xi, eta = sp.symbols("x y xi eta", real=True)
mu1, mu2, sigma1, sigma2, zeta, z = sp.symbols("mu1 mu2 sigma1 sigma2 zeta z", real=True)
T = sp.symbols("T", real=True)
# Real and positive variables(real=True, positive=True)
m, k, alpha, beta, gamma, delta, omega, B, g, eps, A, V0, lambda_param, theta = sp.symbols(
"m k alpha beta gamma delta omega B g eps A V0 lambda theta", real=True, positive=True
)
R, Delta, mu, golden_ratio, A_param, B_param, C_param, f_param, g_param, L_z = sp.symbols(
"R Delta mu golden_ratio A_param B_param C_param f_param g_param L_z", real=True, positive=True
)
# =====================================================================
# Utility functions
# =====================================================================
DYNAMIC_VARS = {x, y, xi, eta}
[docs]
def get_parameters(expr, dim):
"""
Extract free parameters from a Hamiltonian expression,
i.e. symbols that are not dynamical variables.
"""
vars_set = {x, xi} if dim == 1 else {x, y, xi, eta}
return sorted(expr.free_symbols - vars_set, key=lambda s: s.name)
import re # add at the top of the file if not already present
class _InitialConditionError(ValueError):
"""Raised when an initial condition leads to a singular or non-finite Hamiltonian flow."""
pass
def _check_initial_condition_1d(H_num, x0, xi0):
"""
Check that H and its first-order derivatives are finite at (x0, xi0).
Raises _InitialConditionError with a descriptive message if not.
"""
f_H = sp.lambdify((x, xi), H_num, 'numpy')
f_Hx = sp.lambdify((x, xi), sp.diff(H_num, x), 'numpy')
f_Hxi = sp.lambdify((x, xi), sp.diff(H_num, xi), 'numpy')
for label, func in [("H", f_H), ("∂H/∂x", f_Hx), ("∂H/∂ξ", f_Hxi)]:
try:
val = float(func(x0, xi0))
if not np.isfinite(val):
raise _InitialConditionError(
f"{label}({x0}, {xi0}) = {val} (non-finite). "
f"Please choose a different starting point."
)
except (ZeroDivisionError, ValueError, FloatingPointError) as e:
raise _InitialConditionError(
f"{label} is singular at (x0={x0}, xi0={xi0}): {e}"
)
def _check_initial_condition_2d(H_num, x0, y0, xi0, eta0):
"""
Check that H and its first-order derivatives are finite at (x0, y0, xi0, eta0).
Raises _InitialConditionError with a descriptive message if not.
"""
f_H = sp.lambdify((x, y, xi, eta), H_num, 'numpy')
f_Hx = sp.lambdify((x, y, xi, eta), sp.diff(H_num, x), 'numpy')
f_Hy = sp.lambdify((x, y, xi, eta), sp.diff(H_num, y), 'numpy')
f_Hxi = sp.lambdify((x, y, xi, eta), sp.diff(H_num, xi), 'numpy')
f_Heta = sp.lambdify((x, y, xi, eta), sp.diff(H_num, eta), 'numpy')
for label, func in [
("H", f_H), ("∂H/∂x", f_Hx), ("∂H/∂y", f_Hy),
("∂H/∂ξ", f_Hxi), ("∂H/∂η", f_Heta)
]:
try:
val = float(func(x0, y0, xi0, eta0))
if not np.isfinite(val):
raise _InitialConditionError(
f"{label}({x0}, {y0}, {xi0}, {eta0}) = {val} (non-finite). "
f"Please choose a different starting point."
)
except (ZeroDivisionError, ValueError, FloatingPointError) as e:
raise _InitialConditionError(
f"{label} is singular at (x0={x0}, y0={y0}, xi0={xi0}, eta0={eta0}): {e}"
)
[docs]
def visualize_hamiltonian(name: str):
"""
Interactive visualization of a Hamiltonian from the catalog.
Prompts the user to provide:
- numerical values for all free parameters (m, k, eps, ...)
- variable ranges (x_range, xi_range, and y_range/eta_range in 2D)
- initial conditions for geodesics
- E_range, hbar, resolution
Then calls geometry.visualize_symbol (1D) or geometry.visualize_symbol_2d (2D).
Parameters
----------
name : str
Key of the Hamiltonian in the catalog.
Returns
-------
Result of visualize_symbol or visualize_symbol_2d.
Example
-------
>>> visualize_hamiltonian('harmonic_oscillator')
>>> visualize_hamiltonian('henon_heiles')
"""
from geometry import visualize_symbol, visualize_symbol_2d
import matplotlib.pyplot as plt
# ── 1. Retrieve the Hamiltonian ──────────────────────────────────────────
if name not in CATALOG:
raise KeyError(f"Unknown Hamiltonian: '{name}'. "
f"Use list_hamiltonians() to see available entries.")
info = CATALOG[name]
H_expr = info["expr"]
dim = info["dim"]
print(f"\n{'='*60}")
print(f" Hamiltonian : {name} ({dim}D — {info['category']})")
print(f" {info['description']}")
print(f" H = {H_expr}")
print(f"{'='*60}\n")
# ── 2. Assign numerical values to free parameters ────────────────────────
params = get_parameters(H_expr, dim)
param_values = {}
if params:
print("Free parameters detected:", [str(p) for p in params])
print("Enter a numerical value for each parameter.\n")
for p in params:
while True:
try:
val = float(input(f" {p} = "))
param_values[p] = val
break
except ValueError:
print(" ✗ Invalid value, please try again.")
H_num = H_expr.subs(param_values)
else:
print(" (No free parameters — H depends only on dynamical variables)\n")
H_num = H_expr
print(f"\n Substituted H = {H_num}\n")
# ── Helper: ask for a (min, max) range ───────────────────────────────────
def ask_range(label, default=(-3.0, 3.0)):
default_str = f"[{default[0]}, {default[1]}]"
raw = input(f" {label} (min max, default {default_str}) : ").strip()
if not raw:
return default
parts = raw.split()
if len(parts) == 2:
return (float(parts[0]), float(parts[1]))
raise ValueError(f"Expected format: 'min max', got: '{raw}'")
def ask_float(label, default):
raw = input(f" {label} (default {default}) : ").strip()
return float(raw) if raw else default
def ask_int(label, default):
raw = input(f" {label} (default {default}) : ").strip()
return int(raw) if raw else default
# ── 3. Variable ranges ───────────────────────────────────────────────────
print("─── Variable ranges ────────────────────────────────────────────")
x_range = ask_range("x_range ")
xi_range = ask_range("xi_range")
if dim == 2:
y_range = ask_range("y_range ")
eta_range = ask_range("eta_range")
# ── 4. Spectral parameters ───────────────────────────────────────────────
print("\n─── Spectral parameters ────────────────────────────────────────")
use_erange = input(" Define E_range? (y/n, default n) : ").strip().lower()
if use_erange == 'y':
E_range = ask_range("E_range", default=(0.5, 4.0))
else:
E_range = None
hbar = ask_float("hbar ", default=1.0)
resolution = ask_int ("resolution", default=80 if dim == 1 else 40)
# ── 5. Geodesic initial conditions ───────────────────────────────────────
print("\n─── Geodesic initial conditions ────────────────────────────────")
if dim == 1:
print(" Format : x0 xi0 t_max [color]")
print(" Example: 0.0 1.5 10 royalblue")
else:
print(" Format : x0 y0 xi0 eta0 t_max [color]")
print(" Example: 1.0 0.0 0.0 1.5 6.28 royalblue")
# Warn the user if H has known singularities (sqrt, log, 1/x ...)
H_str = str(H_num)
singularity_warnings = []
if 'sqrt' in H_str:
singularity_warnings.append("sqrt(·) — avoid initial conditions where the argument is zero or negative")
if 'log' in H_str:
singularity_warnings.append("log(·) — avoid initial conditions where the argument is zero or negative")
if re.search(r'1/x|1/y|\bx\b\s*\*\*\s*-|\by\b\s*\*\*\s*-', H_str):
singularity_warnings.append("1/x or 1/y — avoid x=0 or y=0")
if singularity_warnings:
print("\n ⚠ Singularity warning for this Hamiltonian:")
for w in singularity_warnings:
print(f" · {w}")
print()
COLORS = ['royalblue', 'crimson', 'seagreen', 'darkorange',
'purple', 'gold', 'teal', 'hotpink']
geodesics_params = []
geo_idx = 0
while True:
raw = input(f" Geodesic {geo_idx+1} (empty line to finish) : ").strip()
if not raw:
if geo_idx == 0:
print(" ✗ At least one geodesic is required.")
continue
break
parts = raw.split()
try:
if dim == 1:
if len(parts) < 3:
raise ValueError
x0, xi0, t_max = float(parts[0]), float(parts[1]), float(parts[2])
color = parts[3] if len(parts) >= 4 else COLORS[geo_idx % len(COLORS)]
# Validate: check H_num and its x-derivative are finite at (x0, xi0)
_check_initial_condition_1d(H_num, x0, xi0)
geodesics_params.append((x0, xi0, t_max, color))
else:
if len(parts) < 5:
raise ValueError
x0, y0 = float(parts[0]), float(parts[1])
xi0, eta0, t_max = float(parts[2]), float(parts[3]), float(parts[4])
color = parts[5] if len(parts) >= 6 else COLORS[geo_idx % len(COLORS)]
# Validate: check H_num and its derivatives are finite at (x0, y0, xi0, eta0)
_check_initial_condition_2d(H_num, x0, y0, xi0, eta0)
geodesics_params.append((x0, y0, xi0, eta0, t_max, color))
geo_idx += 1
except _InitialConditionError as e:
print(f" ✗ Invalid initial condition: {e}")
except (ValueError, IndexError):
expected = "x0 xi0 t_max [color]" if dim == 1 else "x0 y0 xi0 eta0 t_max [color]"
print(f" ✗ Invalid format. Expected: {expected}")
# ── 6. Summary before launching ──────────────────────────────────────────
print(f"\n{'─'*60}")
print(f" Launching visualization of '{name}'...")
if param_values:
print(f" Parameters : { {str(k): v for k, v in param_values.items()} }")
print(f" x_range={x_range}, xi_range={xi_range}", end="")
if dim == 2:
print(f", y_range={y_range}, eta_range={eta_range}", end="")
print(f"\n hbar={hbar}, resolution={resolution}, E_range={E_range}")
print(f" Geodesics : {len(geodesics_params)}")
print(f"{'─'*60}\n")
# ── 7. Call geometry ─────────────────────────────────────────────────────
if dim == 1:
result = visualize_symbol(
symbol = H_num,
x_range = x_range,
xi_range = xi_range,
geodesics_params = geodesics_params,
E_range = E_range,
hbar = hbar,
resolution = resolution,
x_sym = x,
xi_sym = xi,
)
else:
result = visualize_symbol_2d(
symbol = H_num,
x_range = x_range,
y_range = y_range,
xi_range = xi_range,
eta_range = eta_range,
geodesics_params = geodesics_params,
E_range = E_range,
hbar = hbar,
resolution = resolution,
x_sym = x,
y_sym = y,
xi_sym = xi,
eta_sym = eta,
)
plt.show()
return result
[docs]
def get_operator(name: str, param_values: dict = None, quantization: str = 'weyl'):
"""
Instantiate a PseudoDifferentialOperator from a catalog Hamiltonian.
Parameters
----------
name : str
Catalog key.
param_values : dict, optional
Free parameter substitutions, e.g. {'m': 1.0, 'k': 2.0}.
If None, the symbol is used as-is (symbolic parameters kept).
quantization : {'weyl', 'kn'}
Quantization scheme passed to PseudoDifferentialOperator.
Returns
-------
PseudoDifferentialOperator
"""
from psiop import PseudoDifferentialOperator
H, vars, params, info = get_hamiltonian(name)
H_num = H.subs(_resolve_param_values(param_values)) if param_values else H
vars_x = [x] if info['dim'] == 1 else [x, y]
return PseudoDifferentialOperator(expr=H_num, vars_x=vars_x,
mode='symbol', quantization=quantization)
def _resolve_param_values(param_values: dict) -> dict:
"""
Convert string keys in param_values to sympy Symbols,
looking them up in the global SYMBOLS table first to preserve
assumptions (real=True, positive=True, etc.).
"""
if not param_values:
return {}
# Build a lookup from name → sympy Symbol for all known symbols
all_symbols = {s.name: s for s in [
x, y, xi, eta, m, k, alpha, beta, gamma, delta, omega,
B, g, eps, A, V0, lambda_param, theta, R, Delta, mu,
golden_ratio, A_param, B_param, C_param, f_param, g_param,
L_z, mu1, mu2, sigma1, sigma2, zeta, z, T
]}
resolved = {}
for key, val in param_values.items():
if isinstance(key, str):
sym = all_symbols.get(key, sp.Symbol(key))
else:
sym = key # already a sympy Symbol
resolved[sym] = val
return resolved
[docs]
def analyze_hamiltonian(name: str, param_values: dict = None,
x_grid=None, xi_grid=None):
"""
Symbolic and numerical ΨDO analysis of a catalog Hamiltonian.
Computes: operator order, ellipticity, self-adjointness,
principal symbol, formal adjoint, symplectic flow equations.
Returns
-------
dict with keys: 'order', 'is_elliptic', 'is_self_adjoint',
'principal_symbol', 'formal_adjoint', 'symplectic_flow'
"""
import numpy as np
H, vars, params, info = get_hamiltonian(name)
dim = info['dim']
# Substitute parameters BEFORE instantiating the operator
print("param_values = ", param_values)
if param_values:
H_num = H.subs(_resolve_param_values(param_values))
else:
H_num = H
print("H_num = ", H_num)
# Warn if free parameters remain — is_elliptic_numerically will fail
remaining = get_parameters(H_num, dim)
if remaining:
raise ValueError(
f"Cannot run analyze_hamiltonian('{name}'): symbolic parameters remain "
f"after substitution: {[str(p) for p in remaining]}.\n"
f"Please provide numerical values via param_values={{{', '.join(repr(str(p))+': ?' for p in remaining)}}}."
)
# Build operator from the fully numerical symbol
from psiop import PseudoDifferentialOperator
vars_x = [x] if dim == 1 else [x, y]
op = PseudoDifferentialOperator(expr=H_num, vars_x=vars_x,
mode='symbol', quantization='weyl')
if x_grid is None:
x_grid = np.linspace(-3, 3, 50)
if xi_grid is None:
xi_grid = np.linspace(-3, 3, 50)
result = {
'order': op.symbol_order(),
'principal_symbol': op.principal_symbol(),
'formal_adjoint': op.formal_adjoint(),
'symplectic_flow': op.symplectic_flow(),
'is_self_adjoint': op.is_self_adjoint(),
'is_elliptic': op.is_elliptic_numerically(x_grid, xi_grid),
}
return result
[docs]
def trace_hamiltonian(name: str, param_values: dict = None,
numerical: bool = False, x_bounds=None, xi_bounds=None):
"""
Compute the semiclassical trace of a catalog Hamiltonian.
Wraps PseudoDifferentialOperator.trace_formula().
Parameters
----------
x_bounds : tuple (min, max), optional
Spatial integration bounds, e.g. (-3, 3).
xi_bounds : tuple (min, max), optional
Frequency integration bounds, e.g. (-3, 3).
"""
op = get_operator(name, param_values)
# psiop.trace_formula expects x_bounds as ((min, max),) in 1D — wrap accordingly
if x_bounds is not None and op.dim == 1:
x_bounds_wrapped = (x_bounds,)
xi_bounds_wrapped = (xi_bounds,)
else:
x_bounds_wrapped = x_bounds
xi_bounds_wrapped = xi_bounds
return op.trace_formula(
numerical=numerical,
x_bounds=x_bounds_wrapped,
xi_bounds=xi_bounds_wrapped,
)
[docs]
def interactive_hamiltonian(name: str, param_values: dict = None, **kwargs):
"""
Launch the interactive ipywidgets dashboard for a catalog Hamiltonian.
Wraps PseudoDifferentialOperator.interactive_symbol_analysis().
Requires a Jupyter environment.
"""
from psiop import PseudoDifferentialOperator
op = get_operator(name, param_values)
PseudoDifferentialOperator.interactive_symbol_analysis(op, **kwargs)
[docs]
def get_hamiltonian(name: str):
"""
Return Hamiltonian expression, variables, and metadata.
Parameters
----------
name : str
Key identifier for the Hamiltonian.
Returns
-------
H : sympy.Expr
The Hamiltonian expression.
vars : tuple
Variables (x, xi) for 1D or (x, y, xi, eta) for 2D.
params : tuple
Parameters in the Hamiltonian
info : dict
Metadata including dimension, category, and description.
Example
-------
>>> H, vars, params, meta = get_hamiltonian("henon_heiles")
>>> print(meta["description"])
Hénon–Heiles: benchmark for mixed regular/chaotic motion.
"""
if name not in CATALOG:
available = list(CATALOG.keys())[:10]
raise KeyError(
f"Unknown Hamiltonian '{name}'.\n"
f"Available (first 10): {available}\n"
f"Use list_hamiltonians() to see all {len(CATALOG)} entries."
)
info = CATALOG[name]
H = info["expr"]
dim = info["dim"]
vars = (x, xi) if dim == 1 else (x, y, xi, eta)
params = tuple(get_parameters(H, dim))
return H, vars, params, info
[docs]
def list_categories():
"""
List all categories and their counts.
Returns
-------
dict
Dictionary mapping category names to counts.
"""
c = Counter([v["category"] for v in CATALOG.values()])
return dict(c)
[docs]
def list_hamiltonians(category=None, dim=None):
"""
List Hamiltonian names, optionally filtered by category or dimension.
Parameters
----------
category : str, optional
Filter by category (e.g., 'chaotic', 'integrable').
dim : int, optional
Filter by dimension (1 or 2).
Returns
-------
list
List of Hamiltonian names matching the criteria.
Example
-------
>>> list_hamiltonians(category='chaotic')
['henon_heiles', 'quartic_coupled', ...]
>>> list_hamiltonians(dim=1)
['free_particle', 'harmonic_oscillator', ...]
"""
result = []
for name, info in CATALOG.items():
if category and info["category"] != category:
continue
if dim and info["dim"] != dim:
continue
result.append(name)
return sorted(result)
[docs]
def search_hamiltonians(keyword: str):
"""
Search for Hamiltonians by keyword in name or description.
Parameters
----------
keyword : str
Search term (case-insensitive).
Returns
-------
list
List of matching Hamiltonian names.
Example
-------
>>> search_hamiltonians('pendulum')
['double_pendulum_reduced', 'driven_pendulum', 'spherical_pendulum', ...]
"""
keyword = keyword.lower()
result = []
for name, info in CATALOG.items():
if keyword in name.lower() or keyword in info["description"].lower():
result.append(name)
return sorted(result)
[docs]
def print_hamiltonian_info(name: str):
"""
Print detailed information about a specific Hamiltonian.
Parameters
----------
name : str
Hamiltonian identifier.
"""
H, vars, params, info = get_hamiltonian(name)
print(f"\n{'='*70}")
print(f"Hamiltonian: {name}")
print(f"{'='*70}")
print(f"Category: {info['category']}")
print(f"Dimension: {info['dim']}D")
print(f"Variables: {vars}")
print(f"Parameters: {params if params else '(none)'}") # ← nouveau
print(f"\nDescription:\n {info['description']}")
print(f"\nExpression:\n H = {H}")
print(f"{'='*70}\n")
[docs]
def get_catalog_summary():
"""
Return a formatted summary of the entire catalog.
Returns
-------
str
Multi-line summary with statistics.
Note: use `print(get_catalog_summary())`
"""
total = len(CATALOG)
categories = list_categories()
dim_1 = len([v for v in CATALOG.values() if v["dim"] == 1])
dim_2 = len([v for v in CATALOG.values() if v["dim"] == 2])
summary = [
"=" * 70,
"HAMILTONIAN CATALOG SUMMARY",
"=" * 70,
f"Total Hamiltonians: {total}",
f" - 1D systems: {dim_1}",
f" - 2D systems: {dim_2}",
"",
"Categories:",
]
for cat, count in sorted(categories.items(), key=lambda x: -x[1]):
summary.append(f" {cat:20s} : {count:3d}")
summary.append("=" * 70)
return "\n".join(summary)
[docs]
def get_hamiltonians_by_keywords(*keywords):
"""
Multi-keyword search with AND operator.
Example
-------
>>> get_hamiltonians_by_keywords('quantum', 'oscillator')
"""
results = []
for name, info in CATALOG.items():
text = (name + ' ' + info['description']).lower()
if all(kw.lower() in text for kw in keywords):
results.append(name)
return sorted(results)
[docs]
def get_tree():
"""
Returns a hierarchical tree of categories reflecting the full scope
of the extended Hamiltonian catalog (including physical, biological,
social, cognitive, and speculative systems).
Returns
-------
dict
Tree structured by super-categories mapping to subcategories
that actually appear in the catalog, with counts of Hamiltonians per subcategory.
"""
# First, collect all actual categories used and count Hamiltonians per category
from collections import Counter
category_counts = Counter(v["category"] for v in CATALOG.values())
all_categories = set(category_counts.keys())
tree = {
# ────────────────────────────────
# PHYSICAL SCIENCES
# ────────────────────────────────
"Classical & Celestial Mechanics": [
"integrable", "chaotic", "nonlinear", "classical", "integrable_advanced",
"astrophysics", "geophysics", "climate"
],
"Quantum & Atomic Physics": [
"quantum", "atomic", "molecular", "nuclear", "ultracold", "mesoscopic",
"quantum_topological_extended", "semiclassical"
],
"Field Theory & High-Energy Physics": [
"qft", "particle_physics", "string_theory", "quantum_gravity",
"supersymmetry", "bsm", "dark_sector", "neutrino", "exotic_matter",
"field_theory"
],
"Condensed Matter & Materials": [
"lattice", "spin_systems", "spin_glass", "defects", "metamaterials",
"topological", "quantum_info", "quantum_info_advanced", "polymers"
],
"Electromagnetism & Optics": [
"magnetic", "optical", "plasma", "cavity_qed", "nonlinear_optics",
"acoustics", "rotating"
],
"Relativity & Gravitation": [
"relativistic", "black_holes", "cosmology", "geometric", "geometric_advanced"
],
"Statistical & Non-Equilibrium Physics": [
"statistical", "stochastic", "stochastic_advanced", "dissipative",
"non_equilibrium", "reaction_diffusion", "turbulence"
],
"Fluids, Soft Matter & Active Systems": [
"fluid", "granular", "active_matter", "elasticity"
],
"Solitons & Nonlinear Waves": [
"continuum_solitons", "multi_scale_chaos"
],
# ────────────────────────────────
# APPLIED & INTERDISCIPLINARY
# ────────────────────────────────
"Biophysics & Life Sciences": [
"biophysics", "neuroscience", "epidemiology", "public_health", "ecology"
],
"Engineering & Technology": [
"accelerator", "control_theory", "optics", "acoustics"
],
"Earth & Environmental Systems": [
"geophysics", "climate", "agriculture", "urban"
],
# ────────────────────────────────
# INFORMATION, COGNITION & SOCIETY
# ────────────────────────────────
"Information & Computation": [
"quantum_info", "quantum_info_advanced", "symbolic", "optimization",
"inference", "network_dynamics", "modern_extensions"
],
"Cognitive & Psychological Dynamics": [
"cognitive", "wellness", "dream", "neuroscience", "education"
],
"Social & Cultural Systems": [
"econophysics", "game_dynamics", "linguistics", "religion", "folklore",
"digital_culture", "legal", "sports", "urban", "quant_finance"
],
# ────────────────────────────────
# CREATIVE & AESTHETIC DOMAINS
# ────────────────────────────────
"Aesthetic & Design Domains": [
"art_music", "typography", "architecture", "perfumery", "fashion",
"cuisine", "culinary", "generative", "dance", "poetics"
],
"Cultural & Symbolic Practices": [
"ceremony", "mythopoetics", "folklore"
],
# ────────────────────────────────
# WELLNESS & LIFESTYLE
# ────────────────────────────────
"Health & Wellness": [
"wellness", "dream", "gardening"
],
# ────────────────────────────────
# PURE & ADVANCED MATHEMATICS
# ────────────────────────────────
"Mathematical Structures": [
"pure_math", "mathematical", "twistor", "tft", "symmetry_reduced",
"exotic"
],
# ────────────────────────────────
# SPECULATIVE & FRONTIER DOMAINS
# ────────────────────────────────
"Metaphysical & Speculative": [
"metaphysical", "exotic"
],
}
# Replace each subcategory with its actual Hamiltonian count, if present
result = {}
for super_cat, subcats in tree.items():
filtered = {
cat: category_counts[cat]
for cat in subcats
if cat in all_categories and category_counts[cat] > 0
}
if filtered:
result[super_cat] = filtered
return result
[docs]
def export_latex_table(category=None, filename='hamiltonians.tex'):
"""
Exports a LaTeX table of Hamiltonians.
Parameters
----------
category : str, optional
Category to export (all if None).
filename : str
Output file name.
"""
import sympy as sp
# List of Hamiltonians to export
hamiltonians = list_hamiltonians(category=category) if category else list(CATALOG.keys())
# LaTeX table header
lines = [
r"\begin{longtable}{|l|c|p{8cm}|}",
r"\hline",
r"\textbf{Name} & \textbf{Dim} & \textbf{Hamiltonian} \\",
r"\hline",
r"\endfirsthead",
r"\hline",
r"\textbf{Name} & \textbf{Dim} & \textbf{Hamiltonian} \\",
r"\hline",
r"\endhead",
r"\hline",
r"\endfoot",
]
# Add rows for each Hamiltonian
for name in hamiltonians:
info = CATALOG[name]
H_latex = sp.latex(info['expr'])
dim = info['dim']
name_latex = name.replace('_', r'\_')
lines.append(f"{name_latex} & {dim}D & ${H_latex}$" + r" \\" + "\n")
lines.append(r"\hline")
# End of the table
lines.append(r"\end{longtable}")
# Write to file
try:
with open(filename, 'w') as f:
f.write('\n'.join(lines))
print(f"Exported {len(hamiltonians)} Hamiltonians to {filename}")
except IOError as e:
print(f"Error writing to file {filename}: {e}")
[docs]
def get_dimensional_analysis(name: str):
"""
Basic dimensional analysis of a Hamiltonian.
Parameters
----------
name : str
Name of the Hamiltonian.
Returns
-------
dict
Information about structural properties.
"""
H, vars, params, info = get_hamiltonian(name)
terms = H.as_ordered_terms()
analysis = {
'name': name,
'dimension': info['dim'],
'num_terms': len(terms),
'polynomial_degree': 0,
'has_trigonometric': False,
'has_exponential': False,
'has_logarithm': False,
'has_sqrt': False,
'has_abs': False,
'has_rational': False,
'complexity_score': 0
}
H_str = str(H)
analysis['has_trigonometric'] = any(f in H_str for f in ['sin', 'cos', 'tan', 'cot', 'sec', 'csc'])
analysis['has_exponential'] = 'exp' in H_str or '**' in H_str # crude but effective
analysis['has_logarithm'] = 'log' in H_str
analysis['has_sqrt'] = 'sqrt' in H_str
analysis['has_abs'] = 'Abs' in H_str or 'abs(' in H_str.lower()
analysis['has_rational'] = any(op in H_str for op in ['/ ', '/(', '/x', '/y'])
# Safely estimate polynomial degree
for var in vars:
for term in terms:
if term.has(var):
try:
deg = sp.degree(term, var)
if deg is not None and deg >= 0:
analysis['polynomial_degree'] = max(analysis['polynomial_degree'], int(deg))
except (sp.PolynomialError, ValueError, TypeError, AttributeError):
# Non-polynomial term (e.g., log, sqrt, exp) — skip degree calculation
continue
complexity = len(H_str)
complexity += 10 * analysis['num_terms']
complexity += 20 * int(analysis['has_trigonometric'])
complexity += 20 * int(analysis['has_exponential'])
complexity += 15 * int(analysis['has_logarithm'])
complexity += 10 * int(analysis['has_sqrt'])
complexity += 10 * int(analysis['has_abs'])
analysis['complexity_score'] = complexity
return analysis
[docs]
def find_similar_hamiltonians(name: str, top_n=5):
"""
Finds similar Hamiltonians by structural analysis.
Parameters
----------
name : str
Name of the reference Hamiltonian.
top_n : int
Number of results to return.
Returns
-------
list
List of tuples (name, similarity score).
"""
ref_analysis = get_dimensional_analysis(name)
ref_info = CATALOG[name]
similarities = []
for other_name in CATALOG:
if other_name == name:
continue
other_analysis = get_dimensional_analysis(other_name)
other_info = CATALOG[other_name]
score = 0
if ref_info['dim'] == other_info['dim']:
score += 30
if ref_info['category'] == other_info['category']:
score += 40
if ref_analysis['has_trigonometric'] == other_analysis['has_trigonometric']:
score += 10
if ref_analysis['has_exponential'] == other_analysis['has_exponential']:
score += 10
if ref_analysis['has_logarithm'] == other_analysis['has_logarithm']:
score += 10
term_diff = abs(ref_analysis['num_terms'] - other_analysis['num_terms'])
score += max(0, 10 - term_diff)
similarities.append((other_name, score))
similarities.sort(key=lambda x: x[1], reverse=True)
return similarities[:top_n]
[docs]
def validate_hamiltonians():
"""
Validates all Hamiltonians to detect common errors.
Returns
-------
dict
Validation report with warnings and errors.
"""
report = {
'valid': [],
'warnings': [],
'errors': [],
'suspicious': []
}
for name, info in CATALOG.items():
H = info['expr']
H_str = str(H)
issues = []
if 'Derivative' in H_str and 'Derivative(x, x)' in H_str:
issues.append("Contains Derivative(x,x) which equals 1")
if not any(var in H_str for var in ['x', 'y', 'xi', 'eta']):
issues.append("No dynamical variables found")
if ('1/x' in H_str or '1/y' in H_str) and 'eps' not in H_str:
issues.append("Division by coordinate without regularization")
if 'sqrt' in H_str and '-' in H_str:
issues.append("Potential sqrt of negative quantity")
if 'log' in H_str and not 'Abs' in H_str and 'eps' not in H_str:
issues.append("Logarithm without absolute value or regularization")
if len(H_str) > 500:
issues.append(f"Very complex expression (length: {len(H_str)})")
if issues:
report['warnings'].append({
'name': name,
'issues': issues,
'expression': H_str[:100] + ' ...' if len(H_str) > 100 else H_str
})
else:
report['valid'].append(name)
return report
[docs]
def batch_export_hamiltonians(output_dir='hamiltonians_export', formats=['json', 'yaml', 'csv']):
"""
Exports the entire catalog in multiple formats.
Parameters
----------
output_dir : str
Output directory.
formats : list
Desired formats: 'json', 'yaml', 'csv', 'markdown'.
"""
os.makedirs(output_dir, exist_ok=True)
if 'json' in formats:
catalog_json = {}
for name, info in CATALOG.items():
catalog_json[name] = {
'expression': str(info['expr']),
'dimension': info['dim'],
'category': info['category'],
'description': info['description']
}
with open(f'{output_dir}/catalog.json', 'w') as f:
json.dump(catalog_json, f, indent=2)
print(f"✓ Exported to {output_dir}/catalog.json")
if 'yaml' in formats:
try:
import yaml
with open(f'{output_dir}/catalog.yaml', 'w') as f:
yaml.dump(catalog_json, f, default_flow_style=False)
print(f"✓ Exported to {output_dir}/catalog.yaml")
except ImportError:
print("✗ YAML export requires PyYAML package")
if 'csv' in formats:
with open(f'{output_dir}/catalog.csv', 'w') as f:
f.write("Name,Dimension,Category,Description\n")
for name, info in CATALOG.items():
desc = info['description'].replace(',', ';')
f.write(f'{name},{info["dim"]},{info["category"]},"{desc}"\n')
print(f"✓ Exported to {output_dir}/catalog.csv")
if 'markdown' in formats:
with open(f'{output_dir}/catalog.md', 'w') as f:
f.write("# Hamiltonian Catalog\n\n")
f.write(f"**Total Systems**: {len(CATALOG)}\n\n")
for category in sorted(set(info['category'] for info in CATALOG.values())):
f.write(f"\n## {category.replace('_', ' ').title()}\n\n")
hamiltonians = [name for name, info in CATALOG.items() if info['category'] == category]
for name in sorted(hamiltonians):
info = CATALOG[name]
f.write(f"### {name}\n")
f.write(f"- **Dimension**: {info['dim']}D\n")
f.write(f"- **Description**: {info['description']}\n")
f.write(f"- **Expression**: `{info['expr']}`\n\n")
print(f"✓ Exported to {output_dir}/catalog.md")
# ========================================================== #
# N.B. : This list of Hamiltonians was generated using LLMs. #
# ========================================================== #
# =====================================================================
# 1. INTEGRABLE / POLYNOMIAL SYSTEMS
# =====================================================================
H_INTEGRABLE = {
"free_particle": {
"expr": xi**2 / (2*m),
"dim": 1,
"category": "integrable",
"description": "Free particle — straight trajectories, trivial flow.",
},
"harmonic_oscillator": {
"expr": xi**2/(2*m) + k*x**2/2,
"dim": 1,
"category": "integrable",
"description": "1D harmonic oscillator, closed circular trajectories.",
},
"anharmonic_oscillator": {
"expr": xi**2/(2*m) + alpha*x**4/4,
"dim": 1,
"category": "integrable",
"description": "Quartic oscillator — stiffer potential, still integrable.",
},
"double_well": {
"expr": xi**2/(2*m) + alpha*(x**2 - 1)**2,
"dim": 1,
"category": "integrable",
"description": "Symmetric double-well with two minima — tunneling prototype.",
},
"kepler": {
"expr": (xi**2 + eta**2)/(2*m) - k/sp.sqrt(x**2 + y**2 + eps),
"dim": 2,
"category": "integrable",
"description": "Kepler problem: inverse-square central potential (elliptic orbits).",
},
"isotropic_oscillator": {
"expr": (xi**2 + eta**2)/(2*m) + k*(x**2 + y**2)/2,
"dim": 2,
"category": "integrable",
"description": "2D isotropic oscillator — circular orbits, conserved angular momentum.",
},
"anisotropic_oscillator": {
"expr": (xi**2 + eta**2)/(2*m) + 0.5*(x**2 + 2*y**2),
"dim": 2,
"category": "integrable",
"description": "Anisotropic oscillator with rational frequency ratio — Lissajous figures.",
},
"mexican_hat": {
"expr": (xi**2 + eta**2)/2 + (x**2 + y**2 - 1)**2,
"dim": 2,
"category": "integrable",
"description": "Mexican-hat potential — ring of stable equilibria.",
},
"toda_pair": {
"expr": xi**2/2 + alpha*sp.exp(-(x - y)),
"dim": 2,
"category": "integrable",
"description": "Two-particle Toda lattice — exponential repulsion.",
},
"calogero_moser": {
"expr": (xi**2 + eta**2)/2 + g/((x - y)**2 + eps),
"dim": 2,
"category": "integrable",
"description": "Calogero–Moser model with inverse-square interaction.",
},
"sextic_oscillator": {
"expr": xi**2/(2*m) + alpha*x**6/6,
"dim": 1,
"category": "integrable",
"description": "Sextic potential — higher-order polynomial confinement.",
},
"linear_potential": {
"expr": xi**2/(2*m) + g*x,
"dim": 1,
"category": "integrable",
"description": "Particle in constant force field (gravity).",
},
"cubic_potential": {
"expr": xi**2/(2*m) + alpha*x**3/3,
"dim": 1,
"category": "integrable",
"description": "Cubic potential — asymmetric unbounded system.",
},
"quartic_2d": {
"expr": (xi**2 + eta**2)/(2*m) + alpha*(x**4 + y**4)/4,
"dim": 2,
"category": "integrable",
"description": "Separable 2D quartic oscillator.",
},
"radial_power": {
"expr": (xi**2 + eta**2)/(2*m) + alpha*(x**2 + y**2)**2,
"dim": 2,
"category": "integrable",
"description": "Radially symmetric quartic potential.",
},
}
# =====================================================================
# 2. NONLINEAR & CHAOTIC SYSTEMS
# =====================================================================
H_CHAOTIC = {
"henon_heiles": {
"expr": (xi**2 + eta**2)/2 + (x**2 + y**2)/2 + alpha*(x**2*y - y**3/3),
"dim": 2,
"category": "chaotic",
"description": "Hénon–Heiles: benchmark for mixed regular/chaotic motion.",
},
"quartic_coupled": {
"expr": (xi**2 + eta**2)/2 + 0.25*(x**4 + y**4 + alpha*x**2*y**2),
"dim": 2,
"category": "chaotic",
"description": "Quartic coupled oscillator — chaotic for large coupling α.",
},
"double_pendulum_reduced": {
"expr": (xi**2 + eta**2 + xi*eta*sp.cos(x-y))/(2*(1 + sp.sin(x-y)**2)) + (sp.cos(x) + sp.cos(y)),
"dim": 2,
"category": "chaotic",
"description": "Reduced double pendulum — strongly nonlinear and chaotic.",
},
"duffing": {
"expr": xi**2/2 + 0.5*x**2 + 0.25*beta*x**4,
"dim": 1,
"category": "nonlinear",
"description": "Duffing oscillator — bistable potential, nonlinear dynamics.",
},
"driven_pendulum": {
"expr": xi**2/2 + (1 - sp.cos(x)) + alpha*x*sp.cos(omega),
"dim": 1,
"category": "chaotic",
"description": "Driven pendulum — time-dependent forcing, chaotic response.",
},
"standard_map_like": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.cos(x)*sp.cos(y),
"dim": 2,
"category": "chaotic",
"description": "Continuous analogue of the standard map — separatrix chaos.",
},
"quartic_mixed": {
"expr": xi**2/2 + 0.25*x**4 - x,
"dim": 1,
"category": "nonlinear",
"description": "Asymmetric quartic potential — metastability and bifurcation.",
},
"hill_potential": {
"expr": xi**2/2 - alpha*x**2 + beta*x**4,
"dim": 1,
"category": "nonlinear",
"description": "Hill potential — used in celestial and accelerator dynamics.",
},
"henon_heiles_variant": {
"expr": (xi**2 + eta**2)/2 + 0.5*(x**2 + y**2) + alpha*x**2*y - beta*y**3,
"dim": 2,
"category": "chaotic",
"description": "Modified Hénon–Heiles with adjustable nonlinearity.",
},
"yang_mills_reduced": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 - y**2)**2,
"dim": 2,
"category": "chaotic",
"description": "Reduced Yang–Mills model — gauge theory analog.",
},
"stadium_billiard_smooth": {
"expr": (xi**2 + eta**2)/2 + V0/(1 + sp.exp(-alpha*(sp.sqrt(x**2 + y**2) - 1))),
"dim": 2,
"category": "chaotic",
"description": "Smoothed stadium billiard potential — chaotic scattering.",
},
"sinai_billiard_smooth": {
"expr": (xi**2 + eta**2)/2 + V0*sp.exp(-alpha*(x**2 + y**2)),
"dim": 2,
"category": "chaotic",
"description": "Smooth approximation to Sinai billiard with circular scatterer.",
},
"poincare_surface": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.sin(x)*sp.sin(y),
"dim": 2,
"category": "chaotic",
"description": "Poincaré surface of section model — periodic modulation.",
},
"coupled_morse": {
"expr": xi**2/2 + eta**2/2 + alpha*(sp.exp(-2*x) - 2*sp.exp(-x)) + beta*(sp.exp(-2*y) - 2*sp.exp(-y)) + gamma*x*y,
"dim": 2,
"category": "chaotic",
"description": "Coupled Morse oscillators — molecular vibrational chaos.",
},
"van_der_pol": {
"expr": xi**2/2 + y**2/2 + alpha*(x**2 - 1)*x*xi,
"dim": 2,
"category": "nonlinear",
"description": "Van der Pol oscillator — limit cycle dynamics.",
},
}
# =====================================================================
# 3. MAGNETIC & ROTATING SYSTEMS
# =====================================================================
H_MAGNETIC = {
"landau_levels": {
"expr": ((xi - B*y/2)**2 + (eta + B*x/2)**2)/(2*m),
"dim": 2,
"category": "magnetic",
"description": "Charged particle in uniform B field — Landau quantization.",
},
"fock_darwin": {
"expr": ((xi - B*y/2)**2 + (eta + B*x/2)**2)/(2*m) + 0.5*k*(x**2 + y**2),
"dim": 2,
"category": "magnetic",
"description": "Oscillator + magnetic field — Fock–Darwin states.",
},
"coriolis": {
"expr": 0.5*(xi**2 + eta**2) - omega*(x*eta - y*xi),
"dim": 2,
"category": "rotating",
"description": "Coriolis Hamiltonian — dynamics in a rotating frame.",
},
"charged_potential": {
"expr": ((xi - A*y)**2 + (eta + A*x)**2)/2 + alpha*(x**2 + y**2),
"dim": 2,
"category": "magnetic",
"description": "Generic magnetic oscillator with vector potential.",
},
"aharonov_bohm": {
"expr": (xi**2 + eta**2)/(2*m) + (A/(x**2 + y**2 + eps))**2,
"dim": 2,
"category": "magnetic",
"description": "Aharonov–Bohm effect — topological phase from magnetic flux.",
},
"hall_effect": {
"expr": ((xi - B*y)**2 + eta**2)/(2*m) + V0*x,
"dim": 2,
"category": "magnetic",
"description": "Hall effect geometry — drift in crossed E and B fields.",
},
"cyclotron_resonance": {
"expr": ((xi - omega*y)**2 + (eta + omega*x)**2)/(2*m) + alpha*sp.cos(omega),
"dim": 2,
"category": "magnetic",
"description": "Cyclotron resonance with time-periodic drive.",
},
"penning_trap": {
"expr": ((xi - B*y/2)**2 + (eta + B*x/2)**2)/(2*m) + alpha*(x**2 + y**2 - 2*x**2),
"dim": 2,
"category": "magnetic",
"description": "Penning trap — quadrupole electric + magnetic confinement.",
},
"magnetic_bottle": {
"expr": (xi**2 + eta**2)/(2*m) + B*(1 + alpha*x**2)*(x**2 + y**2)/2,
"dim": 2,
"category": "magnetic",
"description": "Magnetic bottle trap — inhomogeneous field confinement.",
},
}
# =====================================================================
# 4. OPTICAL / REFRACTIVE SYSTEMS
# =====================================================================
H_OPTICAL = {
"graded_index": {
"expr": (xi**2 + eta**2)/(2*(1 + alpha*(x**2 + y**2))),
"dim": 2,
"category": "optical",
"description": "Gradient-index fiber — geodesics bend toward the axis.",
},
"photonic_crystal": {
"expr": (xi**2 + eta**2)/2 + alpha*(sp.cos(x) + sp.cos(y)),
"dim": 2,
"category": "optical",
"description": "Periodic photonic lattice — band-structure analog.",
},
"anisotropic_medium": {
"expr": 0.5*((1+alpha*x**2)*xi**2 + (1+beta*y**2)*eta**2),
"dim": 2,
"category": "optical",
"description": "Anisotropic refractive medium — direction-dependent propagation.",
},
"waveguide_bent": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2 + beta*(y - 0.2*x**2)**2,
"dim": 2,
"category": "optical",
"description": "Bent optical waveguide — model for ray focusing.",
},
"kerr_nonlinearity": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2)**2,
"dim": 2,
"category": "optical",
"description": "Kerr nonlinearity — self-focusing in optical media.",
},
"bragg_grating": {
"expr": xi**2/(2*m) + V0*sp.cos(2*alpha*x),
"dim": 1,
"category": "optical",
"description": "Bragg grating — periodic refractive index modulation.",
},
"fiber_coupler": {
"expr": (xi**2 + eta**2)/2 + 0.5*(x**2 + y**2) + alpha*x*y,
"dim": 2,
"category": "optical",
"description": "Optical fiber coupler — evanescent wave coupling.",
},
"soliton_potential": {
"expr": xi**2/2 - V0/sp.cosh(alpha*x)**2,
"dim": 1,
"category": "optical",
"description": "Soliton potential — nonlinear wave localization.",
},
"photonic_waveguide_array": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x) + sp.cos(y) + alpha*sp.cos(x + y)),
"dim": 2,
"category": "optical",
"description": "Coupled waveguide array with diagonal coupling.",
},
}
# =====================================================================
# 5. RELATIVISTIC & SEMICLASSICAL
# =====================================================================
H_RELATIVISTIC = {
"relativistic_free": {
"expr": sp.sqrt(xi**2 + eta**2 + m**2),
"dim": 2,
"category": "relativistic",
"description": "Relativistic free particle, energy–momentum relation.",
},
"klein_gordon": {
"expr": sp.sqrt(xi**2 + m**2) + 0.5*k*x**2,
"dim": 1,
"category": "relativistic",
"description": "Klein–Gordon Hamiltonian with harmonic confinement.",
},
"dirac_radial": {
"expr": sp.sqrt(xi**2 + (alpha*x)**2 + m**2),
"dim": 1,
"category": "relativistic",
"description": "1D Dirac-type dispersion with position-dependent mass term.",
},
"semi_classical": {
"expr": xi**2/2 + alpha*x**4/4 + eps*xi**4,
"dim": 1,
"category": "semiclassical",
"description": "Schrödinger-like with small semiclassical correction in ξ.",
},
"relativistic_oscillator": {
"expr": sp.sqrt(xi**2 + eta**2 + m**2) + k*(x**2 + y**2)/2,
"dim": 2,
"category": "relativistic",
"description": "Relativistic harmonic oscillator.",
},
"dirac_coulomb": {
"expr": sp.sqrt(xi**2 + m**2) - alpha/sp.sqrt(x**2 + eps),
"dim": 1,
"category": "relativistic",
"description": "Dirac equation with Coulomb potential.",
},
"relativistic_kepler": {
"expr": sp.sqrt(xi**2 + eta**2 + m**2) - k/sp.sqrt(x**2 + y**2 + eps),
"dim": 2,
"category": "relativistic",
"description": "Relativistic Kepler problem — perihelion precession.",
},
}
# =====================================================================
# 6. ATOMIC / MOLECULAR POTENTIALS
# =====================================================================
H_POTENTIALS = {
"morse": {
"expr": xi**2/(2*m) + alpha*(sp.exp(-2*x) - 2*sp.exp(-x)),
"dim": 1,
"category": "atomic",
"description": "Morse potential — bound vibrational states, dissociation limit.",
},
"yukawa": {
"expr": xi**2/2 - g*sp.exp(-alpha*sp.sqrt(x**2 + y**2))/(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "molecular",
"description": "Yukawa potential — screened Coulomb interaction.",
},
"lennard_jones": {
"expr": xi**2/2 + 4*alpha*((beta/x)**12 - (beta/x)**6),
"dim": 1,
"category": "molecular",
"description": "Lennard–Jones — molecular bonding and repulsion.",
},
"gaussian_barrier": {
"expr": xi**2/2 + alpha*sp.exp(-x**2),
"dim": 1,
"category": "scattering",
"description": "Gaussian barrier — simple tunneling benchmark.",
},
"coulomb_2d": {
"expr": (xi**2 + eta**2)/2 - 1/sp.sqrt(x**2 + y**2 + eps),
"dim": 2,
"category": "atomic",
"description": "2D Coulomb potential — hydrogen-like bound states.",
},
"poschl_teller": {
"expr": xi**2/(2*m) - V0/sp.cosh(alpha*x)**2,
"dim": 1,
"category": "atomic",
"description": "Pöschl–Teller potential — exactly solvable quantum well.",
},
"eckart_barrier": {
"expr": xi**2/(2*m) + V0/(sp.cosh(alpha*x)**2),
"dim": 1,
"category": "scattering",
"description": "Eckart barrier — tunneling and reflection coefficient.",
},
"rosen_morse": {
"expr": xi**2/(2*m) - V0*sp.tanh(alpha*x) + V0,
"dim": 1,
"category": "atomic",
"description": "Rosen–Morse potential — asymmetric molecular interaction.",
},
"woods_saxon": {
"expr": xi**2/(2*m) - V0/(1 + sp.exp((x - beta)/alpha)),
"dim": 1,
"category": "nuclear",
"description": "Woods–Saxon potential — nuclear mean field approximation.",
},
"hulthen": {
"expr": xi**2/(2*m) - alpha*sp.exp(-x)/(1 - sp.exp(-x)),
"dim": 1,
"category": "atomic",
"description": "Hulthén potential — screened Coulomb for atomic screening.",
},
"manning_rosen": {
"expr": xi**2/(2*m) - V0/sp.sinh(alpha*x)**2,
"dim": 1,
"category": "molecular",
"description": "Manning–Rosen potential — molecular bond model.",
},
"buckingham": {
"expr": xi**2/(2*m) + alpha*sp.exp(-beta*x) - gamma/x**6,
"dim": 1,
"category": "molecular",
"description": "Buckingham potential — exp-6 molecular interaction.",
},
}
# =====================================================================
# 7. GEOMETRIC & FLUID-INSPIRED FLOWS
# =====================================================================
H_GEOMETRIC = {
"geodesic_plane": {
"expr": (xi**2 + eta**2)/(2*(1 + alpha*(x**2 + y**2))),
"dim": 2,
"category": "geometric",
"description": "Geodesic flow on curved metric g=(1+αr²) — defocusing curvature.",
},
"vortex_pair": {
"expr": -sp.log(sp.sqrt((x-y)**2 + eps)),
"dim": 2,
"category": "fluid",
"description": "Simplified vortex interaction energy in 2D Euler flow.",
},
"shallow_water": {
"expr": xi**2/2 + g*x,
"dim": 1,
"category": "fluid",
"description": "Reduced shallow-water Hamiltonian — slope-induced motion.",
},
"magnetic_geodesic": {
"expr": ((xi - B*y)**2 + (eta + B*x)**2)/2,
"dim": 2,
"category": "geometric",
"description": "Geodesic flow under magnetic field (twisted symplectic form).",
},
"schwarzschild_radial": {
"expr": (1 - 2*m/x)*xi**2/2 + alpha**2/(2*x**2),
"dim": 1,
"category": "geometric",
"description": "Schwarzschild radial geodesic — general relativity orbit.",
},
"hyperbolic_geodesic": {
"expr": (xi**2 + eta**2)/(2*y**2),
"dim": 2,
"category": "geometric",
"description": "Geodesic flow on hyperbolic plane (Poincaré half-plane).",
},
"point_vortex_3": {
"expr": -sp.log(sp.sqrt(x**2 + y**2 + eps)) - sp.log(sp.sqrt((x-1)**2 + y**2 + eps)),
"dim": 2,
"category": "fluid",
"description": "Three-vortex interaction in 2D ideal fluid.",
},
"rossby_wave": {
"expr": (xi**2 + eta**2)/2 + beta*x*y,
"dim": 2,
"category": "fluid",
"description": "Rossby wave Hamiltonian — atmospheric/oceanic dynamics.",
},
}
# =====================================================================
# 8. QUANTUM & CONDENSED MATTER SYSTEMS
# =====================================================================
H_QUANTUM = {
"harmonic_spin_orbit": {
"expr": (xi**2 + eta**2)/(2*m) + k*(x**2 + y**2)/2 + alpha*(x*eta - y*xi),
"dim": 2,
"category": "quantum",
"description": "Harmonic oscillator with spin-orbit coupling.",
},
"rashba_hamiltonian": {
"expr": (xi**2 + eta**2)/(2*m) + alpha*(x*eta - y*xi) + beta*(x**2 + y**2),
"dim": 2,
"category": "quantum",
"description": "Rashba spin-orbit interaction in 2DEG.",
},
"jaynes_cummings": {
"expr": omega*xi + alpha*(x*xi + x**2),
"dim": 1,
"category": "quantum",
"description": "Jaynes–Cummings model — atom-cavity interaction.",
},
"hofstadter_butterfly": {
"expr": (xi**2 + eta**2)/2 + alpha*(sp.cos(x) + sp.cos(y)) - B*(x*eta - y*xi),
"dim": 2,
"category": "quantum",
"description": "Hofstadter model — fractal energy spectrum in magnetic field.",
},
"bose_hubbard_continuum": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x) + sp.cos(y)) + alpha*(x**2 + y**2)**2,
"dim": 2,
"category": "quantum",
"description": "Continuum limit of Bose–Hubbard model.",
},
"superconducting_pairing": {
"expr": (xi**2 + eta**2)/(2*m) - delta*(x*y + xi*eta),
"dim": 2,
"category": "quantum",
"description": "BCS pairing Hamiltonian — superconductivity.",
},
"gross_pitaevskii": {
"expr": xi**2/(2*m) + V0*sp.sin(x)**2 + g*x**2,
"dim": 1,
"category": "quantum",
"description": "Gross–Pitaevskii equation for BEC — mean field theory.",
},
}
# =====================================================================
# 9. ASTROPHYSICAL & GRAVITATIONAL SYSTEMS
# =====================================================================
H_ASTROPHYSICS = {
"schwarzschild_orbit": {
"expr": (xi**2 + eta**2)/(2*m) - k/sp.sqrt(x**2 + y**2) + alpha/(x**2 + y**2),
"dim": 2,
"category": "astrophysics",
"description": "Schwarzschild effective potential — GR corrections to orbits.",
},
"three_body_restricted": {
"expr": (xi**2 + eta**2)/2 - 1/sp.sqrt((x+alpha)**2 + y**2 + eps) - alpha/sp.sqrt((x-1)**2 + y**2 + eps),
"dim": 2,
"category": "astrophysics",
"description": "Restricted three-body problem — Lagrange points.",
},
"tidal_force": {
"expr": xi**2/(2*m) + k*x**2/2 - alpha*x**3,
"dim": 1,
"category": "astrophysics",
"description": "Tidal force approximation near massive body.",
},
"galactic_rotation": {
"expr": (xi**2 + eta**2)/(2*m) - k/sp.sqrt(x**2 + y**2 + eps) + omega*(x*eta - y*xi),
"dim": 2,
"category": "astrophysics",
"description": "Galactic disk rotation with dark matter halo.",
},
"planetary_ring": {
"expr": (xi**2 + eta**2)/2 - 1/sp.sqrt(x**2 + y**2 + eps) + omega**2*(x**2 + y**2)/2,
"dim": 2,
"category": "astrophysics",
"description": "Particle dynamics in planetary ring system.",
},
}
# =====================================================================
# 10. LATTICE & PERIODIC SYSTEMS
# =====================================================================
H_LATTICE = {
"kronig_penney": {
"expr": xi**2/(2*m) + V0*sum([sp.DiracDelta(x - n) for n in range(-3, 4)]),
"dim": 1,
"category": "lattice",
"description": "Kronig–Penney model — periodic delta potentials.",
},
"mathieu": {
"expr": xi**2/(2*m) + V0*sp.cos(2*x),
"dim": 1,
"category": "lattice",
"description": "Mathieu equation — parametric resonance in periodic systems.",
},
"tight_binding": {
"expr": (xi**2 + eta**2)/2 + 2*V0*(sp.cos(x) + sp.cos(y)),
"dim": 2,
"category": "lattice",
"description": "Tight-binding approximation on square lattice.",
},
"harper_model": {
"expr": xi**2/2 + 2*V0*sp.cos(x + alpha*y),
"dim": 2,
"category": "lattice",
"description": "Harper model — quasiperiodic potential, fractal spectrum.",
},
"aubry_andre": {
"expr": xi**2/(2*m) + V0*sp.cos(2*sp.pi*alpha*x),
"dim": 1,
"category": "lattice",
"description": "Aubry–André model — metal-insulator transition.",
},
"wannier_stark": {
"expr": xi**2/(2*m) + V0*sp.cos(x) + g*x,
"dim": 1,
"category": "lattice",
"description": "Wannier–Stark ladder — Bloch oscillations in tilted lattice.",
},
"kagome_lattice": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x) + sp.cos(y) + sp.cos(x-y)),
"dim": 2,
"category": "lattice",
"description": "Kagome lattice geometry — frustrated magnetic systems.",
},
"hexagonal_lattice": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x) + sp.cos(y) + sp.cos(x+y)),
"dim": 2,
"category": "lattice",
"description": "Hexagonal (graphene-like) lattice structure.",
},
}
# =====================================================================
# 11. STOCHASTIC & DISSIPATIVE SYSTEMS
# =====================================================================
H_DISSIPATIVE = {
"damped_oscillator": {
"expr": xi**2/(2*m) + k*x**2/2 + gamma*x*xi,
"dim": 1,
"category": "dissipative",
"description": "Damped harmonic oscillator — energy dissipation.",
},
"caldirola_kanai": {
"expr": sp.exp(-gamma)*xi**2/(2*m) + sp.exp(gamma)*k*x**2/2,
"dim": 1,
"category": "dissipative",
"description": "Caldirola–Kanai Hamiltonian — time-dependent damping.",
},
"rayleigh_oscillator": {
"expr": xi**2/2 + x**2/2 + alpha*xi*(xi**2 - 1),
"dim": 1,
"category": "dissipative",
"description": "Rayleigh oscillator — nonlinear damping model.",
},
"fokker_planck": {
"expr": xi**2/2 - gamma*sp.log(1 + x**2),
"dim": 1,
"category": "dissipative",
"description": "Fokker–Planck effective Hamiltonian.",
},
}
# =====================================================================
# 12. BIOPHYSICAL & CHEMICAL SYSTEMS
# =====================================================================
H_BIOPHYSICS = {
"protein_folding": {
"expr": xi**2/(2*m) + alpha*(1 - sp.cos(x))**2 + beta*(1 - sp.cos(y))**2 + gamma*sp.sin(x)*sp.sin(y),
"dim": 2,
"category": "biophysics",
"description": "Simplified protein dihedral angle dynamics.",
},
"dna_twist": {
"expr": xi**2/(2*m) + k*x**2/2 + alpha*sp.cos(beta*x),
"dim": 1,
"category": "biophysics",
"description": "DNA torsional dynamics — supercoiling model.",
},
"michaelis_menten": {
"expr": xi**2/2 + V0*x/(k + x),
"dim": 1,
"category": "biophysics",
"description": "Michaelis–Menten enzyme kinetics effective potential.",
},
"hodgkin_huxley_reduced": {
"expr": xi**2/2 + alpha*x**3 - beta*x,
"dim": 1,
"category": "biophysics",
"description": "Reduced Hodgkin–Huxley — neural spike dynamics.",
},
"fitzhugh_nagumo": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**3/3 - x) + y,
"dim": 2,
"category": "biophysics",
"description": "FitzHugh–Nagumo model — excitable media.",
},
"chemotaxis": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.log(1 + x**2 + y**2),
"dim": 2,
"category": "biophysics",
"description": "Chemotactic cell migration — logarithmic attraction.",
},
}
# =====================================================================
# 13. PLASMA & ELECTROMAGNETIC SYSTEMS
# =====================================================================
H_PLASMA = {
"plasma_wave": {
"expr": (xi**2 + eta**2)/2 + omega**2*(x**2 + y**2)/2 + alpha*x*y,
"dim": 2,
"category": "plasma",
"description": "Plasma oscillation with wave coupling.",
},
"vlasov_poisson": {
"expr": xi**2/2 + alpha*sp.sin(x),
"dim": 1,
"category": "plasma",
"description": "Vlasov–Poisson system — plasma collective effects.",
},
"debye_shielding": {
"expr": xi**2/2 - g*sp.exp(-alpha*sp.sqrt(x**2 + eps))/(sp.sqrt(x**2 + eps)),
"dim": 1,
"category": "plasma",
"description": "Debye-screened Coulomb potential in plasma.",
},
"tokamak_particle": {
"expr": ((xi - B*y)**2 + eta**2)/2 + V0*(1 - sp.cos(x)),
"dim": 2,
"category": "plasma",
"description": "Charged particle in tokamak — toroidal confinement.",
},
"cyclotron_maser": {
"expr": ((xi - omega*y)**2 + (eta + omega*x)**2)/2 + alpha*sp.cos(x),
"dim": 2,
"category": "plasma",
"description": "Cyclotron maser instability — wave-particle resonance.",
},
}
# =====================================================================
# 14. ACCELERATOR & BEAM PHYSICS
# =====================================================================
H_ACCELERATOR = {
"rf_cavity": {
"expr": xi**2/(2*m) + V0*sp.sin(omega*x),
"dim": 1,
"category": "accelerator",
"description": "RF cavity acceleration — synchrotron motion.",
},
"betatron_oscillation": {
"expr": xi**2/(2*m) + k*(1 + alpha*sp.cos(y))*x**2/2,
"dim": 2,
"category": "accelerator",
"description": "Betatron oscillations in circular accelerator.",
},
"synchrotron_radiation": {
"expr": xi**2/(2*m) - gamma*xi + k*x**2/2,
"dim": 1,
"category": "accelerator",
"description": "Energy loss from synchrotron radiation.",
},
"space_charge": {
"expr": (xi**2 + eta**2)/(2*m) + alpha*sp.log(x**2 + y**2 + eps),
"dim": 2,
"category": "accelerator",
"description": "Space charge effects in particle beams.",
},
"chromaticity": {
"expr": xi**2/(2*m) + k*(1 + alpha*xi)*x**2/2,
"dim": 1,
"category": "accelerator",
"description": "Chromatic aberration in beam optics.",
},
}
# =====================================================================
# 15. EXOTIC & ADVANCED SYSTEMS
# =====================================================================
H_EXOTIC = {
"fractal_potential": {
"expr": xi**2/2 + V0*sp.sin(x)*sp.sin(alpha*x)*sp.sin(alpha**2*x),
"dim": 1,
"category": "exotic",
"description": "Multi-scale fractal potential — self-similar structure.",
},
"random_matrix": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 - y**2) + beta*x*y,
"dim": 2,
"category": "exotic",
"description": "Random matrix ensemble Hamiltonian.",
},
"supersymmetric_qm": {
"expr": xi**2/2 + (sp.diff(V0*sp.tanh(x), x))**2 - sp.diff(sp.diff(V0*sp.tanh(x), x), x),
"dim": 1,
"category": "exotic",
"description": "Supersymmetric quantum mechanics partner potential.",
},
"pt_symmetric": {
"expr": xi**2/(2*m) + sp.I*V0*x**3,
"dim": 1,
"category": "exotic",
"description": "PT-symmetric (non-Hermitian) Hamiltonian.",
},
"anyonic_oscillator": {
"expr": (xi**2 + eta**2)/2 + k*(x**2 + y**2)/2 + alpha*(x*eta - y*xi)**2,
"dim": 2,
"category": "exotic",
"description": "Anyon oscillator — fractional statistics.",
},
"noncommutative_space": {
"expr": (xi**2 + eta**2)/2 + k*(x**2 + y**2)/2 + theta*(x*eta - y*xi),
"dim": 2,
"category": "exotic",
"description": "Noncommutative geometry — quantum space structure.",
},
"q_deformed": {
"expr": (sp.exp(xi) - sp.exp(-xi))/(2*sp.sinh(alpha)) + k*x**2/2,
"dim": 1,
"category": "exotic",
"description": "q-deformed oscillator — quantum group symmetry.",
},
"levy_flight": {
"expr": sp.Abs(xi)**alpha + V0*x**2/2,
"dim": 1,
"category": "exotic",
"description": "Lévy flight dynamics — superdiffusion anomalous transport.",
},
}
# =====================================================================
# 16. ADDITIONAL CLASSICAL MECHANICS
# =====================================================================
H_CLASSICAL_EXTENDED = {
"spherical_pendulum": {
"expr": (xi**2 + eta**2)/(2*m) + m*g*(1 - sp.cos(x))*sp.sin(y)**2,
"dim": 2,
"category": "classical",
"description": "Spherical pendulum — 3D pendulum motion.",
},
"spinning_top": {
"expr": (xi**2 + eta**2)/(2*m) + omega*(x*eta - y*xi) + m*g*x,
"dim": 2,
"category": "classical",
"description": "Spinning top (simplified) — precession dynamics.",
},
"wilberforce_spring": {
"expr": xi**2/(2*m) + eta**2/(2*m) + k*x**2/2 + k*y**2/2 + alpha*x*y,
"dim": 2,
"category": "classical",
"description": "Wilberforce pendulum — coupled translation-rotation.",
},
"atwood_machine": {
"expr": xi**2/(2*m) - m*g*x,
"dim": 1,
"category": "classical",
"description": "Atwood machine — constrained pulley system.",
},
"coupled_pendula": {
"expr": (xi**2 + eta**2)/(2*m) + g*(sp.cos(x) + sp.cos(y)) + k*(x - y)**2/2,
"dim": 2,
"category": "classical",
"description": "Two coupled pendula — normal mode analysis.",
},
"roller_coaster": {
"expr": xi**2/(2*m) + m*g*sp.sin(alpha*x),
"dim": 1,
"category": "classical",
"description": "Roller coaster dynamics — gravity on curved track.",
},
"brachistochrone": {
"expr": sp.sqrt(1 + sp.diff(y, x)**2)*sp.sqrt(2*g*y),
"dim": 1,
"category": "classical",
"description": "Brachistochrone problem — fastest descent curve.",
},
}
# =====================================================================
# 17. TOPOLOGICAL & GAUGE SYSTEMS
# =====================================================================
H_TOPOLOGICAL = {
"chern_insulator": {
"expr": (xi**2 + eta**2)/2 + alpha*(sp.cos(x) + sp.cos(y)) + beta*sp.sin(x)*sp.sin(y),
"dim": 2,
"category": "topological",
"description": "Chern insulator — topological band structure.",
},
"haldane_model": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x) + sp.cos(y) + sp.cos(x+y)) + alpha*sp.sin(x)*sp.sin(y),
"dim": 2,
"category": "topological",
"description": "Haldane model — quantum Hall effect without Landau levels.",
},
"su_schrieffer_heeger": {
"expr": xi**2/(2*m) + V0*(sp.cos(x) + alpha*sp.cos(x/2)),
"dim": 1,
"category": "topological",
"description": "SSH model — topological edge states.",
},
"berry_phase": {
"expr": (xi**2 + eta**2)/2 + omega*(x*eta - y*xi) + V0*sp.cos(sp.atan2(y, x)),
"dim": 2,
"category": "topological",
"description": "System with geometric Berry phase.",
},
"monopole_field": {
"expr": (xi**2 + eta**2)/(2*m) + g*sp.atan2(y, x),
"dim": 2,
"category": "topological",
"description": "Dirac magnetic monopole — topological magnetic charge.",
},
}
# =====================================================================
# 18. NONLINEAR OPTICS & SOLITONS
# =====================================================================
H_NONLINEAR_OPTICS = {
"nls_cubic": {
"expr": xi**2/(2*m) + alpha*x**4,
"dim": 1,
"category": "nonlinear_optics",
"description": "Cubic nonlinear Schrödinger equation — bright solitons.",
},
"nls_quintic": {
"expr": xi**2/(2*m) + alpha*x**4 - beta*x**6,
"dim": 1,
"category": "nonlinear_optics",
"description": "Quintic NLS — soliton stability and collapse.",
},
"derivative_nls": {
"expr": xi**2/(2*m) + alpha*x**2*xi,
"dim": 1,
"category": "nonlinear_optics",
"description": "Derivative NLS — Alfvén waves, plasma physics.",
},
"davey_stewartson": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2)**2 + beta*x*y,
"dim": 2,
"category": "nonlinear_optics",
"description": "Davey–Stewartson equation — 2D soliton interactions.",
},
"sine_gordon": {
"expr": xi**2/(2*m) + V0*(1 - sp.cos(x)),
"dim": 1,
"category": "nonlinear_optics",
"description": "Sine-Gordon equation — kinks and breathers.",
},
"phi4_theory": {
"expr": xi**2/(2*m) + 0.5*m**2*x**2 + lambda_param*x**4/4,
"dim": 1,
"category": "nonlinear_optics",
"description": "φ⁴ field theory — domain walls in phase transitions.",
},
"ablowitz_ladik": {
"expr": xi**2/2 + alpha*sp.sin(x)/(1 + beta*sp.cos(x)),
"dim": 1,
"category": "nonlinear_optics",
"description": "Ablowitz–Ladik lattice — integrable discrete NLS.",
},
"manakov_system": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**4 + y**4 + 2*x**2*y**2),
"dim": 2,
"category": "nonlinear_optics",
"description": "Manakov system — vector solitons, polarization coupling.",
},
"kadomtsev_petviashvili": {
"expr": (xi**2 + eta**2)/2 + alpha*x**3 + beta*x*y,
"dim": 2,
"category": "nonlinear_optics",
"description": "KP equation — 2D shallow water waves.",
},
}
# =====================================================================
# 19. SPIN SYSTEMS & MAGNETIC MODELS
# =====================================================================
H_SPIN_SYSTEMS = {
"heisenberg_classical": {
"expr": -alpha*(sp.cos(x-y)) - beta*(x**2 + y**2),
"dim": 2,
"category": "spin_systems",
"description": "Classical Heisenberg model — spin exchange interaction.",
},
"ising_transverse": {
"expr": -alpha*x - beta*sp.tanh(x),
"dim": 1,
"category": "spin_systems",
"description": "Transverse field Ising model — quantum phase transition.",
},
"xy_model": {
"expr": -alpha*sp.cos(x - y),
"dim": 2,
"category": "spin_systems",
"description": "XY model — planar spins, Kosterlitz-Thouless transition.",
},
"dzyaloshinskii_moriya": {
"expr": (xi**2 + eta**2)/2 + alpha*(x*eta - y*xi) - beta*sp.cos(x)*sp.cos(y),
"dim": 2,
"category": "spin_systems",
"description": "Dzyaloshinskii–Moriya interaction — chiral magnetism.",
},
"landau_lifshitz": {
"expr": -alpha*x*y - beta*(x**2 + y**2)/2,
"dim": 2,
"category": "spin_systems",
"description": "Landau–Lifshitz equation — magnetization dynamics.",
},
"skyrmion": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - x**2 - y**2)**2 + beta*(x*eta - y*xi),
"dim": 2,
"category": "spin_systems",
"description": "Skyrmion texture — topological magnetic soliton.",
},
}
# =====================================================================
# 20. REACTION-DIFFUSION & PATTERN FORMATION
# =====================================================================
H_REACTION_DIFFUSION = {
"fisher_kpp": {
"expr": xi**2/(2*m) - alpha*x*(1 - x),
"dim": 1,
"category": "reaction_diffusion",
"description": "Fisher–KPP equation — population dynamics, traveling waves.",
},
"allen_cahn": {
"expr": xi**2/(2*m) + alpha*(x**2 - 1)**2,
"dim": 1,
"category": "reaction_diffusion",
"description": "Allen–Cahn equation — phase separation dynamics.",
},
"cahn_hilliard": {
"expr": xi**2/2 + alpha*(x**2 - 1)**2 - beta*xi**2,
"dim": 1,
"category": "reaction_diffusion",
"description": "Cahn–Hilliard equation — spinodal decomposition.",
},
"brusselator": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2*y - beta*x,
"dim": 2,
"category": "reaction_diffusion",
"description": "Brusselator model — chemical oscillations, Turing patterns.",
},
"schnakenberg": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2*y - beta*(x + y),
"dim": 2,
"category": "reaction_diffusion",
"description": "Schnakenberg model — autocatalytic reactions.",
},
"gierer_meinhardt": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2/y - beta*x,
"dim": 2,
"category": "reaction_diffusion",
"description": "Gierer–Meinhardt model — biological morphogenesis.",
},
}
# =====================================================================
# 21. ELASTICITY & CONTINUUM MECHANICS
# =====================================================================
H_ELASTICITY = {
"euler_bernoulli_beam": {
"expr": xi**2/(2*m) + k*x**4/4,
"dim": 1,
"category": "elasticity",
"description": "Euler–Bernoulli beam — bending energy (polynomial approximation).",
},
"timoshenko_beam": {
"expr": xi**2/(2*m) + eta**2/(2*m) + k*x**2/2 + alpha*(x - y)**2,
"dim": 2,
"category": "elasticity",
"description": "Timoshenko beam — coupled bending-shear modes.",
},
"kirchhoff_plate": {
"expr": (xi**2 + eta**2)/2 + k*(x**2 + y**2)**2,
"dim": 2,
"category": "elasticity",
"description": "Kirchhoff plate theory — thin plate bending (simplified).",
},
"neo_hookean": {
"expr": xi**2/(2*m) + alpha*(x**2 + 1/x**2),
"dim": 1,
"category": "elasticity",
"description": "Neo-Hookean material — nonlinear elasticity.",
},
"mooney_rivlin": {
"expr": xi**2/(2*m) + alpha*(x**2 - 1) + beta*(1/x**2 - 1),
"dim": 1,
"category": "elasticity",
"description": "Mooney–Rivlin model — rubber elasticity.",
},
}
# =====================================================================
# 22. INFORMATION THEORY & STATISTICAL MECHANICS
# =====================================================================
H_STATISTICAL = {
"maxwell_boltzmann": {
"expr": xi**2/(2*m) - k*sp.log(1 + sp.exp(-x)),
"dim": 1,
"category": "statistical",
"description": "Maxwell–Boltzmann distribution effective potential.",
},
"fermi_dirac": {
"expr": xi**2/(2*m) - k*sp.log(1 + sp.exp(-x/k)),
"dim": 1,
"category": "statistical",
"description": "Fermi–Dirac statistics — electron gas.",
},
"bose_einstein": {
"expr": xi**2/(2*m) + k*sp.log(1 - sp.exp(-x/k)),
"dim": 1,
"category": "statistical",
"description": "Bose–Einstein condensation effective Hamiltonian.",
},
"ising_mean_field": {
"expr": -alpha*x**2 + beta*x**4,
"dim": 1,
"category": "statistical",
"description": "Ising model mean-field free energy.",
},
"potts_model": {
"expr": -alpha*sp.cos(2*sp.pi*x/3) - beta*sp.cos(4*sp.pi*x/3),
"dim": 1,
"category": "statistical",
"description": "q-state Potts model — generalized Ising.",
},
}
# =====================================================================
# 23. NEUROSCIENCE & NEURAL NETWORKS
# =====================================================================
H_NEUROSCIENCE = {
"hopfield_network": {
"expr": -(xi**2 + eta**2)/2 - alpha*x*y,
"dim": 2,
"category": "neuroscience",
"description": "Hopfield network — associative memory energy.",
},
"wilson_cowan": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.tanh(x) - beta*sp.tanh(y),
"dim": 2,
"category": "neuroscience",
"description": "Wilson–Cowan model — neural population dynamics.",
},
"izhikevich": {
"expr": 0.5*xi**2 + 0.04*x**2 + 5*x - y,
"dim": 2,
"category": "neuroscience",
"description": "Izhikevich neuron — efficient spiking model.",
},
"hindmarsh_rose": {
"expr": (xi**2 + eta**2)/2 + alpha*x**3 - beta*x - y,
"dim": 2,
"category": "neuroscience",
"description": "Hindmarsh–Rose neuron — bursting behavior.",
},
"morris_lecar": {
"expr": xi**2/2 + eta**2/2 + alpha*x*(x - beta)*(x - 1) - y,
"dim": 2,
"category": "neuroscience",
"description": "Morris–Lecar model — barnacle muscle fiber.",
},
}
# =====================================================================
# 24. ECONOPHYSICS & SOCIAL DYNAMICS
# =====================================================================
H_ECONOPHYSICS = {
"black_scholes": {
"expr": xi**2*x**2/(2*m) + alpha*x*xi,
"dim": 1,
"category": "econophysics",
"description": "Black–Scholes as Hamiltonian — option pricing.",
},
"heston_model": {
"expr": xi**2/2 + eta**2/2 + alpha*y*(x**2 - beta),
"dim": 2,
"category": "econophysics",
"description": "Heston stochastic volatility model.",
},
"ising_market": {
"expr": -alpha*sp.tanh(x)*sp.tanh(y),
"dim": 2,
"category": "econophysics",
"description": "Ising-like market interaction — herding behavior.",
},
"voter_model": {
"expr": -alpha*x*y,
"dim": 2,
"category": "econophysics",
"description": "Voter model — opinion dynamics.",
},
}
# =====================================================================
# 25. QUANTUM FIELD THEORY INSPIRED
# =====================================================================
H_QFT = {
"schwinger_model": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2) + beta*x*y,
"dim": 2,
"category": "qft",
"description": "Schwinger model — QED in 1+1 dimensions.",
},
"thirring_model": {
"expr": xi**2/(2*m) + alpha*x**4,
"dim": 1,
"category": "qft",
"description": "Thirring model — self-interacting fermions.",
},
"gross_neveu": {
"expr": xi**2/(2*m) + alpha*x**2 + beta*x**4,
"dim": 1,
"category": "qft",
"description": "Gross–Neveu model — asymptotic freedom.",
},
"coleman_weinberg": {
"expr": xi**2/(2*m) + alpha*x**4*sp.log(x**2/beta),
"dim": 1,
"category": "qft",
"description": "Coleman–Weinberg potential — radiative corrections.",
},
"higgs_mexican_hat": {
"expr": (xi**2 + eta**2)/2 - alpha*(x**2 + y**2) + beta*(x**2 + y**2)**2,
"dim": 2,
"category": "qft",
"description": "Higgs potential — spontaneous symmetry breaking.",
},
}
# =====================================================================
# 26. ADDITIONAL EXOTIC & MATHEMATICAL
# =====================================================================
H_MATHEMATICAL = {
"painleve_transcendent": {
"expr": xi**2/2 + alpha*x**3 + beta*x,
"dim": 1,
"category": "mathematical",
"description": "Painlevé transcendent — special function dynamics.",
},
"weierstrass": {
"expr": xi**2/(2*m) + alpha*sp.elliptic_k(x),
"dim": 1,
"category": "mathematical",
"description": "Weierstrass elliptic function potential.",
},
# "jacobi_elliptic": {
# "expr": xi**2/(2*m) + alpha*sp.jacobi_sn(x, m)**2,
# "dim": 1,
# "category": "mathematical",
# "description": "Jacobi elliptic function potential.",
# },
"hypergeometric": {
"expr": xi**2/(2*m) + alpha*sp.hyper((1, 1), (2,), x),
"dim": 1,
"category": "mathematical",
"description": "Hypergeometric function potential.",
},
"lambert_w": {
"expr": xi**2/(2*m) + alpha*x*sp.exp(x),
"dim": 1,
"category": "mathematical",
"description": "Lambert W function related potential.",
},
"zeta_potential": {
"expr": xi**2/(2*m) + alpha/x**2,
"dim": 1,
"category": "mathematical",
"description": "Riemann zeta related inverse square potential.",
},
}
# =====================================================================
# 27. COSMOLOGY & EARLY UNIVERSE
# =====================================================================
H_COSMOLOGY = {
"inflaton_chaotic": {
"expr": xi**2/(2*m) + 0.5*m**2*x**2,
"dim": 1,
"category": "cosmology",
"description": "Chaotic inflation — quadratic potential.",
},
"inflaton_starobinsky": {
"expr": xi**2/(2*m) + alpha*(1 - sp.exp(-beta*x)),
"dim": 1,
"category": "cosmology",
"description": "Starobinsky inflation — R² gravity.",
},
"quintessence": {
"expr": xi**2/(2*m) + V0*sp.exp(-alpha*x),
"dim": 1,
"category": "cosmology",
"description": "Quintessence dark energy — exponential potential.",
},
"ekpyrotic": {
"expr": -xi**2/(2*m) + V0*sp.exp(alpha*x),
"dim": 1,
"category": "cosmology",
"description": "Ekpyrotic universe — negative kinetic energy.",
},
}
# =====================================================================
# 28. TURBULENCE & FLUID DYNAMICS AVANCÉE
# =====================================================================
H_TURBULENCE = {
"kolmogorov_flow": {
"expr": (xi**2 + eta**2)/2 + V0*sp.sin(y),
"dim": 2,
"category": "turbulence",
"description": "Kolmogorov flow — forced 2D turbulence.",
},
"rayleigh_benard": {
"expr": (xi**2 + eta**2)/2 + alpha*x*y - beta*y**2,
"dim": 2,
"category": "turbulence",
"description": "Rayleigh–Bénard convection — thermal instability.",
},
"taylor_couette": {
"expr": (xi**2 + eta**2)/2 - omega*(x*eta - y*xi) + alpha*x**2,
"dim": 2,
"category": "turbulence",
"description": "Taylor–Couette flow — rotating cylinders.",
},
"karman_vortex": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x)*sp.sinh(y)),
"dim": 2,
"category": "turbulence",
"description": "von Kármán vortex street — cylinder wake.",
},
"burgers_potential": {
"expr": xi**2/2 + alpha*x**3/3,
"dim": 1,
"category": "turbulence",
"description": "Burgers equation potential — shock formation.",
},
"kdv_equation": {
"expr": xi**2/2 + alpha*x**3,
"dim": 1,
"category": "turbulence",
"description": "Korteweg–de Vries (simplified) — shallow water solitons.",
},
"navier_stokes_2d": {
"expr": (xi**2 + eta**2)/2 + alpha*(x*y)**2,
"dim": 2,
"category": "turbulence",
"description": "2D Navier–Stokes enstrophy (simplified).",
},
"hasegawa_mima": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.log(1 + x**2 + y**2),
"dim": 2,
"category": "turbulence",
"description": "Hasegawa–Mima plasma turbulence.",
},
}
# =====================================================================
# 29. GRANULAR MATTER & SOFT MATTER
# =====================================================================
H_GRANULAR = {
"hertz_contact": {
"expr": xi**2/(2*m) + alpha*sp.Abs(x)**(3/2),
"dim": 1,
"category": "granular",
"description": "Hertzian contact — elastic collision of spheres.",
},
"durian_foam": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.Abs(sp.sqrt(x**2 + y**2) - 1)**(5/2),
"dim": 2,
"category": "granular",
"description": "Durian foam model — soft sphere packing.",
},
"jamming_transition": {
"expr": (xi**2 + eta**2)/2 + V0*sp.Heaviside(1 - sp.sqrt(x**2 + y**2))*sp.Abs(1 - sp.sqrt(x**2 + y**2))**(3/2),
"dim": 2,
"category": "granular",
"description": "Jamming transition — athermal soft spheres.",
},
"frenkel_kontorova": {
"expr": xi**2/(2*m) + k*(x - sp.sin(x))**2/2,
"dim": 1,
"category": "granular",
"description": "Frenkel–Kontorova model — dislocation dynamics.",
},
"stick_slip": {
"expr": xi**2/(2*m) + k*x**2/2 - alpha*sp.sign(xi)*sp.exp(-beta*sp.Abs(xi)),
"dim": 1,
"category": "granular",
"description": "Stick-slip friction — rate-and-state friction.",
},
"sandpile": {
"expr": (xi**2 + eta**2)/2 + g*sp.sqrt(x**2 + y**2),
"dim": 2,
"category": "granular",
"description": "Sandpile model — self-organized criticality.",
},
}
# =====================================================================
# 30. ACTIVE MATTER & LIVING SYSTEMS
# =====================================================================
H_ACTIVE_MATTER = {
"vicsek_model": {
"expr": (xi**2 + eta**2)/2 + alpha*(x*xi + y*eta)/sp.sqrt(x**2 + y**2 + eps),
"dim": 2,
"category": "active_matter",
"description": "Vicsek model — collective motion of self-propelled particles.",
},
"active_brownian": {
"expr": (xi**2 + eta**2)/2 - V0*(x*sp.cos(sp.atan2(eta, xi)) + y*sp.sin(sp.atan2(eta, xi))),
"dim": 2,
"category": "active_matter",
"description": "Active Brownian particle — self-propulsion.",
},
"toner_tu": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2) - beta*(x*xi + y*eta),
"dim": 2,
"category": "active_matter",
"description": "Toner–Tu theory — flocking hydrodynamics.",
},
"bacterial_swarm": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.log(1 + x**2 + y**2) + beta*(x*xi + y*eta),
"dim": 2,
"category": "active_matter",
"description": "Bacterial swarming — chemotaxis and active flow.",
},
"run_and_tumble": {
"expr": xi**2/(2*m) - V0*sp.sign(x) + alpha*sp.exp(-beta*sp.Abs(x)),
"dim": 1,
"category": "active_matter",
"description": "Run-and-tumble dynamics — bacterial locomotion.",
},
}
# =====================================================================
# 31. METAMATERIALS & PHONONIC CRYSTALS
# =====================================================================
H_METAMATERIALS = {
"negative_refraction": {
"expr": -sp.sqrt(xi**2 + eta**2) + V0*(sp.cos(x) + sp.cos(y)),
"dim": 2,
"category": "metamaterials",
"description": "Negative refraction — backward wave propagation.",
},
"dirac_cone": {
"expr": sp.sqrt(xi**2 + eta**2) + V0*(sp.cos(x) + sp.cos(y) + sp.cos(x-y)),
"dim": 2,
"category": "metamaterials",
"description": "Dirac cone dispersion — graphene-like.",
},
"topological_insulator": {
"expr": (xi**2 + eta**2)/2 + alpha*(sp.sin(x)**2 + sp.sin(y)**2) + beta*sp.sin(x)*sp.sin(y),
"dim": 2,
"category": "metamaterials",
"description": "Topological insulator — edge state protection.",
},
"phononic_bandgap": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(2*x) + sp.cos(2*y)),
"dim": 2,
"category": "metamaterials",
"description": "Phononic crystal — elastic wave bandgap.",
},
"pentamode_material": {
"expr": (xi**2 + eta**2 + 2*alpha*xi*eta)/2 + beta*(x**2 + y**2),
"dim": 2,
"category": "metamaterials",
"description": "Pentamode metamaterial — acoustic cloaking.",
},
"hyperbolic_metamaterial": {
"expr": xi**2/(2*m) - eta**2/(2*m) + V0*(x**2 + y**2),
"dim": 2,
"category": "metamaterials",
"description": "Hyperbolic dispersion — extreme anisotropy.",
},
}
# =====================================================================
# 32. QUANTUM COMPUTING & INFORMATION
# =====================================================================
H_QUANTUM_INFO = {
"transverse_ising_chain": {
"expr": -alpha*sp.cos(x) - beta*x,
"dim": 1,
"category": "quantum_info",
"description": "Transverse Ising — quantum phase transition.",
},
"xyz_spin_chain": {
"expr": -alpha*sp.cos(x) - beta*sp.cos(y) - gamma*sp.cos(x-y),
"dim": 2,
"category": "quantum_info",
"description": "XYZ model — fully anisotropic spin chain.",
},
"kitaev_chain": {
"expr": xi**2/(2*m) + delta*sp.cos(x) + alpha*sp.sin(x),
"dim": 1,
"category": "quantum_info",
"description": "Kitaev chain — Majorana fermions.",
},
"toric_code": {
"expr": -(sp.cos(x) + sp.cos(y) + sp.cos(x+y)),
"dim": 2,
"category": "quantum_info",
"description": "Toric code — topological quantum error correction.",
},
"cluster_state": {
"expr": -sp.cos(x)*sp.cos(y),
"dim": 2,
"category": "quantum_info",
"description": "Cluster state — measurement-based quantum computing.",
},
"rydberg_blockade": {
"expr": (xi**2 + eta**2)/(2*m) + V0/(sp.Abs(x-y)**6 + eps),
"dim": 2,
"category": "quantum_info",
"description": "Rydberg blockade — quantum simulation.",
},
}
# =====================================================================
# 33. GEOPHYSICS & PLANETARY SCIENCE
# =====================================================================
H_GEOPHYSICS = {
"seismic_wave": {
"expr": (xi**2 + eta**2)/(2*(1 + alpha*sp.exp(-beta*y))),
"dim": 2,
"category": "geophysics",
"description": "Seismic wave in stratified medium.",
},
"mantle_convection": {
"expr": (xi**2 + eta**2)/2 + alpha*y*(sp.exp(-x**2) - 1),
"dim": 2,
"category": "geophysics",
"description": "Mantle convection — thermal plumes.",
},
"core_oscillation": {
"expr": (xi**2 + eta**2)/2 - g/sp.sqrt(x**2 + y**2 + eps) + omega**2*(x**2 + y**2),
"dim": 2,
"category": "geophysics",
"description": "Earth's core free oscillation.",
},
"tsunami_wave": {
"expr": xi**2/(2*m) + g*sp.sqrt(x),
"dim": 1,
"category": "geophysics",
"description": "Tsunami propagation — shallow water approximation.",
},
"dynamo_effect": {
"expr": (xi**2 + eta**2)/2 + alpha*(x*eta - y*xi) - beta*sp.cos(x)*sp.cos(y),
"dim": 2,
"category": "geophysics",
"description": "Geodynamo — planetary magnetic field generation.",
},
}
# =====================================================================
# 34. CLIMATE & ATMOSPHERIC DYNAMICS
# =====================================================================
H_CLIMATE = {
"lorenz63": {
"expr": (xi**2 + eta**2)/2 + alpha*x*y - beta*y,
"dim": 2,
"category": "climate",
"description": "Lorenz-63 system — deterministic chaos in convection.",
},
"lorenz96": {
"expr": xi**2/2 + (x - sp.sin(2*sp.pi*alpha))**2,
"dim": 1,
"category": "climate",
"description": "Lorenz-96 — atmospheric predictability model.",
},
"hadley_cell": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.sin(y)*x,
"dim": 2,
"category": "climate",
"description": "Hadley cell circulation — tropical convection.",
},
"enso_oscillation": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.tanh(beta*x)*y,
"dim": 2,
"category": "climate",
"description": "El Niño Southern Oscillation — coupled ocean-atmosphere.",
},
"ice_sheet_flow": {
"expr": xi**2/(2*m) + g*x - alpha*x**3,
"dim": 1,
"category": "climate",
"description": "Ice sheet dynamics — plastic flow model.",
},
}
# =====================================================================
# 35. QUANTUM OPTICS & CAVITY QED
# =====================================================================
H_CAVITY_QED = {
"rabi_oscillation": {
"expr": omega*xi/2 + alpha*x*sp.cos(omega),
"dim": 1,
"category": "cavity_qed",
"description": "Rabi oscillations — two-level system in field.",
},
"tavis_cummings": {
"expr": omega*(xi + eta) + alpha*(x + y)*sp.cos(omega),
"dim": 2,
"category": "cavity_qed",
"description": "Tavis–Cummings — multiple atoms in cavity.",
},
"dicke_model": {
"expr": omega*xi + alpha*x*sp.sqrt(xi),
"dim": 1,
"category": "cavity_qed",
"description": "Dicke superradiance — collective atom-light coupling.",
},
"purcell_effect": {
"expr": (xi**2 + eta**2)/2 + omega*(x**2 + y**2)/2 + alpha*x*y,
"dim": 2,
"category": "cavity_qed",
"description": "Purcell effect — cavity-enhanced emission.",
},
"optomechanics": {
"expr": omega*xi + k*x**2/2 + alpha*x*xi,
"dim": 1,
"category": "cavity_qed",
"description": "Cavity optomechanics — light-matter coupling.",
},
}
# =====================================================================
# 36. CRYSTAL DEFECTS & SOLID STATE
# =====================================================================
H_DEFECTS = {
"edge_dislocation": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.atan2(y, x),
"dim": 2,
"category": "defects",
"description": "Edge dislocation — topological defect in crystal.",
},
"screw_dislocation": {
"expr": (xi**2 + eta**2)/2 + alpha*y*sp.log(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "defects",
"description": "Screw dislocation — helical lattice defect.",
},
"vacancy_diffusion": {
"expr": xi**2/(2*m) + V0*sum([sp.exp(-alpha*(x - n)**2) for n in range(-3, 4)]),
"dim": 1,
"category": "defects",
"description": "Vacancy hopping in crystal lattice.",
},
"peierls_nabarro": {
"expr": xi**2/(2*m) + V0*sp.sin(2*sp.pi*x)**2,
"dim": 1,
"category": "defects",
"description": "Peierls–Nabarro potential — dislocation core.",
},
"twin_boundary": {
"expr": xi**2/(2*m) + alpha*sp.tanh(beta*x)**2,
"dim": 1,
"category": "defects",
"description": "Twin boundary — grain boundary energy.",
},
}
# =====================================================================
# 37. ULTRA-COLD ATOMS & BEC
# =====================================================================
H_ULTRACOLD = {
"optical_lattice_bec": {
"expr": xi**2/(2*m) + V0*sp.sin(x)**2 + g*x**2,
"dim": 1,
"category": "ultracold",
"description": "BEC in optical lattice — Bloch oscillations.",
},
"josephson_junction_bec": {
"expr": -alpha*sp.cos(x) + beta*x**2,
"dim": 1,
"category": "ultracold",
"description": "BEC Josephson junction — macroscopic tunneling.",
},
"feshbach_resonance": {
"expr": xi**2/(2*m) + alpha*(x**2 - beta)**2/(x**2 + gamma),
"dim": 1,
"category": "ultracold",
"description": "Feshbach resonance — tunable interactions.",
},
"raman_coupling": {
"expr": (xi**2 + eta**2)/(2*m) + omega*y + alpha*x*y,
"dim": 2,
"category": "ultracold",
"description": "Raman-coupled BEC — spin-orbit coupling.",
},
"vortex_lattice_bec": {
"expr": (xi**2 + eta**2)/2 - omega*(x*eta - y*xi) + alpha*(x**2 + y**2),
"dim": 2,
"category": "ultracold",
"description": "Vortex lattice in rotating BEC.",
},
}
# =====================================================================
# 38. STOCHASTIC PROCESSES & LÉVY FLIGHTS
# =====================================================================
H_STOCHASTIC = {
"ornstein_uhlenbeck": {
"expr": xi**2/(2*m) + alpha*x**2/2 - gamma*x*xi,
"dim": 1,
"category": "stochastic",
"description": "Ornstein–Uhlenbeck process — mean-reverting noise.",
},
"levy_stable": {
"expr": sp.Abs(xi)**alpha/alpha + V0*x**2/2,
"dim": 1,
"category": "stochastic",
"description": "Lévy stable process — heavy-tailed jumps.",
},
"fractional_brownian": {
"expr": sp.Abs(xi)**(2*alpha) + V0*x**2/2,
"dim": 1,
"category": "stochastic",
"description": "Fractional Brownian motion — long-range correlations.",
},
"continuous_time_random_walk": {
"expr": xi**2/(2*m) + V0*sp.exp(-alpha*sp.Abs(x)),
"dim": 1,
"category": "stochastic",
"description": "CTRW — anomalous diffusion.",
},
}
# =====================================================================
# 39. THÉORIE DES CORDES & GRAVITÉ QUANTIQUE
# =====================================================================
H_STRING_THEORY = {
"nambu_goto": {
"expr": sp.sqrt((xi**2 + eta**2) * (1 + alpha*(x - y)**2)),
"dim": 2,
"category": "string_theory",
"description": "Discrete approximation of Nambu–Goto action using two coupled points on the string.",
},
"polyakov": {
"expr": (xi**2 + eta**2)/2 + alpha*(x - y)**2,
"dim": 2,
"category": "string_theory",
"description": "Discretized Polyakov string as coupled oscillators with tension term.",
},
"ads_cft_particle": {
"expr": sp.sqrt(xi**2 + eta**2)/y + V0*y**(-delta),
"dim": 2,
"category": "string_theory",
"description": "Particle in AdS space — holographic correspondence.",
},
"brane_fluctuation": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2) + beta/(x**2 + y**2 + eps),
"dim": 2,
"category": "string_theory",
"description": "D-brane fluctuations — open string endpoints.",
},
"randall_sundrum": {
"expr": sp.exp(-2*alpha*sp.Abs(y))*xi**2/(2*m) + V0*sp.exp(-alpha*sp.Abs(y)),
"dim": 2,
"category": "string_theory",
"description": "Randall–Sundrum warped geometry.",
},
"dvali_gabadadze": {
"expr": (xi**2 + eta**2)/2 - 1/(sp.sqrt(x**2 + y**2) + eps) + alpha*sp.sqrt(x**2 + y**2),
"dim": 2,
"category": "string_theory",
"description": "DGP model — massive gravity modification.",
},
"regge_trajectory": {
"expr": alpha*xi**2 + beta*x**2,
"dim": 1,
"category": "string_theory",
"description": "Regge trajectory — rotating string spectrum.",
},
}
# =====================================================================
# 40. PHYSIQUE DES PARTICULES & QCD
# =====================================================================
H_PARTICLE_PHYSICS = {
"quark_confinement": {
"expr": xi**2/(2*m) + alpha*sp.Abs(x),
"dim": 1,
"category": "particle_physics",
"description": "Linear confinement potential — QCD string.",
},
"cornell_potential": {
"expr": xi**2/(2*m) - alpha/sp.Abs(x) + beta*sp.Abs(x),
"dim": 1,
"category": "particle_physics",
"description": "Cornell potential — quarkonium (charmonium, bottomonium).",
},
"instanton": {
"expr": (xi**2 + eta**2)/2 + V0/(1 + alpha*(x**2 + y**2))**2,
"dim": 2,
"category": "particle_physics",
"description": "Instanton solution — quantum tunneling in QFT.",
},
"sphaleron": {
"expr": (xi**2 + eta**2)/2 + V0*sp.sin(x)**2*sp.sin(y)**2/(sp.sin(x)**2 + sp.sin(y)**2 + eps),
"dim": 2,
"category": "particle_physics",
"description": "Sphaleron — baryon number violation.",
},
"skyrme_model": {
"expr": (xi**2 + eta**2)/2 + alpha*(xi**2 + eta**2) + beta*(x**2 + y**2)**2,
"dim": 2,
"category": "particle_physics",
"description": "Skyrme model (simplified) — topological solitons as baryons.",
},
"electroweak_phase": {
"expr": xi**2/(2*m) - alpha*x**2 + beta*x**4 + gamma*x**3,
"dim": 1,
"category": "particle_physics",
"description": "Electroweak phase transition potential.",
},
"glueball": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.Abs(x*y) + beta*(x**2 + y**2),
"dim": 2,
"category": "particle_physics",
"description": "Glueball — bound state of gluons.",
},
"parton_distribution": {
"expr": xi**2/(2*m) + V0*x**(alpha)*(1-x)**beta,
"dim": 1,
"category": "particle_physics",
"description": "Parton distribution function — deep inelastic scattering.",
},
}
# =====================================================================
# 41. GRAVITÉ QUANTIQUE & LOOP QUANTUM GRAVITY
# =====================================================================
H_QUANTUM_GRAVITY = {
"wheeler_dewitt": {
"expr": xi**2/(2*m) + V0*sp.exp(3*x),
"dim": 1,
"category": "quantum_gravity",
"description": "Wheeler–DeWitt equation — quantum cosmology (minisuperspace).",
},
"ashtekar_variable": {
"expr": (xi**2 + eta**2)/2 + alpha*(x*eta - y*xi)**2,
"dim": 2,
"category": "quantum_gravity",
"description": "Ashtekar variables — loop quantum gravity.",
},
"spin_network": {
"expr": -alpha*sp.sqrt(x*(x+1)) - beta*sp.sqrt(y*(y+1)),
"dim": 2,
"category": "quantum_gravity",
"description": "Spin network dynamics — discrete quantum geometry.",
},
"causal_set": {
"expr": xi**2/(2*m) + sum([V0*sp.Heaviside(x - n) for n in range(-5, 6)]),
"dim": 1,
"category": "quantum_gravity",
"description": "Causal set — discrete spacetime structure.",
},
"horava_lifshitz": {
"expr": xi**2/(2*m) + alpha*xi**4 + beta*x**2,
"dim": 1,
"category": "quantum_gravity",
"description": "Hořava–Lifshitz gravity — anisotropic scaling with higher derivatives.",
},
}
# =====================================================================
# 42. SYSTÈMES INTÉGRABLES CLASSIQUES AVANCÉS
# =====================================================================
H_INTEGRABLE_ADVANCED = {
"kowalevski_top": {
"expr": (xi**2 + 2*eta**2)/2 + m*g*x,
"dim": 2,
"category": "integrable_advanced",
"description": "Kowalevski top — integrable spinning top.",
},
"goryachev_chaplygin": {
"expr": (xi**2 + eta**2 + 4*xi*eta)/2 + alpha*x,
"dim": 2,
"category": "integrable_advanced",
"description": "Goryachev–Chaplygin top — another integrable case.",
},
"garnier_system": {
"expr": (xi**2 + eta**2)/2 + alpha/(x - y) + beta/(x + y),
"dim": 2,
"category": "integrable_advanced",
"description": "Garnier system — higher-order Painlevé.",
},
"schlesinger_system": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.log(sp.Abs(x - y)) + beta*sp.log(sp.Abs(x + y)),
"dim": 2,
"category": "integrable_advanced",
"description": "Schlesinger system — isomonodromic deformations.",
},
"bogoyavlensky_lattice": {
"expr": xi**2/(2*m) + alpha*sp.exp(x - sp.sin(x)),
"dim": 1,
"category": "integrable_advanced",
"description": "Bogoyavlensky–Toda lattice — integrable discretization.",
},
"ruijsenaars_schneider": {
"expr": sp.Product(sp.sinh((xi - eta + k)/2)/sp.sinh(k/2), (k, 1, alpha)),
"dim": 2,
"category": "integrable_advanced",
"description": "Ruijsenaars–Schneider model — relativistic Calogero.",
},
}
# =====================================================================
# 43. SYSTÈMES HORS ÉQUILIBRE & THERMODYNAMIQUE
# =====================================================================
H_NON_EQUILIBRIUM = {
"jarzynski_work": {
"expr": xi**2/(2*m) + k*(x - lambda_param)**2/2,
"dim": 1,
"category": "non_equilibrium",
"description": "Jarzynski equality setup — nonequilibrium work.",
},
"crooks_fluctuation": {
"expr": xi**2/(2*m) + V0*(sp.tanh(alpha*x) + 1),
"dim": 1,
"category": "non_equilibrium",
"description": "Crooks fluctuation theorem — time-reversal asymmetry.",
},
"mpemba_effect": {
"expr": xi**2/(2*m) + alpha*x**2*(1 - sp.exp(-beta*x**2)),
"dim": 1,
"category": "non_equilibrium",
"description": "Mpemba effect model — anomalous cooling.",
},
"heat_engine": {
"expr": xi**2/(2*m) + k*x**2/2 - alpha*x*sp.cos(omega),
"dim": 1,
"category": "non_equilibrium",
"description": "Quantum heat engine — Carnot-like cycle.",
},
"loschmidt_echo": {
"expr": (xi**2 + eta**2)/2 + V0*sp.cos(x)*sp.cos(y) + eps*sp.sin(x)*sp.sin(y),
"dim": 2,
"category": "non_equilibrium",
"description": "Loschmidt echo — quantum irreversibility.",
},
"kibble_zurek": {
"expr": xi**2/(2*m) - alpha*(1 - lambda_param)*x**2 + beta*x**4,
"dim": 1,
"category": "non_equilibrium",
"description": "Kibble–Zurek mechanism — defect formation in phase transitions.",
},
}
# =====================================================================
# 44. THÉORIE DES TWISTEURS & GÉOMÉTRIE COMPLEXE
# =====================================================================
H_TWISTOR = {
"penrose_twistor": {
"expr": (xi**2 + eta**2)/2 + sp.I*alpha*(x*eta - y*xi),
"dim": 2,
"category": "twistor",
"description": "Penrose twistor space — complex spacetime geometry.",
},
"hitchin_system": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.log(sp.Abs(x**2 + y**2)),
"dim": 2,
"category": "twistor",
"description": "Hitchin system — integrable gauge theory.",
},
"calabi_yau_geodesic": {
"expr": (xi**2 + eta**2)/(2*(1 + alpha*(x**2 + y**2))**2),
"dim": 2,
"category": "twistor",
"description": "Geodesic on Calabi–Yau manifold.",
},
"kaehler_potential": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.log(1 + x**2 + y**2),
"dim": 2,
"category": "twistor",
"description": "Kähler geometry — complex differential geometry.",
},
}
# =====================================================================
# 45. SUPERSYMÉTRIE & SUPERGRAVITÉ
# =====================================================================
H_SUPERSYMMETRY = {
"susy_harmonic": {
"expr": xi**2/(2*m) + k*x**2/2 + alpha*xi,
"dim": 1,
"category": "supersymmetry",
"description": "Supersymmetric harmonic oscillator (simplified bosonic sector).",
},
"witten_index": {
"expr": xi**2/(2*m) + (2*V0*x)**2/2 - V0,
"dim": 1,
"category": "supersymmetry",
"description": "Witten index — SUSY partner potential.",
},
"n2_susy": {
"expr": (xi**2 + eta**2)/2 + V0*(x**2 + y**2) + alpha*(xi**2 + eta**2)*(x**2 + y**2),
"dim": 2,
"category": "supersymmetry",
"description": "N=2 supersymmetry — extended SUSY with coupling.",
},
"seiberg_witten": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 - y**2)**2 + beta*x*y,
"dim": 2,
"category": "supersymmetry",
"description": "Seiberg–Witten theory — N=2 gauge theory.",
},
"sugra_scalar": {
"expr": xi**2/(2*m) - alpha*sp.exp(beta*x) + gamma*sp.exp(2*beta*x),
"dim": 1,
"category": "supersymmetry",
"description": "Supergravity scalar potential.",
},
}
# =====================================================================
# 46. MATIÈRE NOIRE & ÉNERGIE NOIRE
# =====================================================================
H_DARK_SECTOR = {
"wimp_scattering": {
"expr": xi**2/(2*m) + g*sp.exp(-alpha*x**2)/x**2,
"dim": 1,
"category": "dark_sector",
"description": "WIMP–nucleon scattering — dark matter detection.",
},
"axion_field": {
"expr": xi**2/(2*m) + V0*(1 - sp.cos(x/alpha)),
"dim": 1,
"category": "dark_sector",
"description": "Axion field — dark matter candidate.",
},
"fuzzy_dark_matter": {
"expr": (xi**2 + eta**2)/(2*m) + g*(x**2 + y**2)**2,
"dim": 2,
"category": "dark_sector",
"description": "Fuzzy dark matter — ultralight bosons.",
},
"chameleon_field": {
"expr": xi**2/(2*m) + V0*sp.exp(alpha*x) + beta/x**4,
"dim": 1,
"category": "dark_sector",
"description": "Chameleon mechanism — screened fifth force.",
},
"dark_photon": {
"expr": (xi**2 + eta**2)/(2*m) - eps*alpha/(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "dark_sector",
"description": "Dark photon — hidden sector U(1).",
},
"phantom_energy": {
"expr": -xi**2/(2*m) + V0*x**2,
"dim": 1,
"category": "dark_sector",
"description": "Phantom dark energy — w < -1 equation of state.",
},
}
# =====================================================================
# 47. PHYSIQUE DES NEUTRINOS
# =====================================================================
H_NEUTRINO = {
"neutrino_oscillation": {
"expr": (xi**2 + eta**2)/(2*m) + alpha*sp.sin(2*sp.atan2(y, x)),
"dim": 2,
"category": "neutrino",
"description": "Neutrino flavor oscillations — mixing matrix.",
},
"msw_effect": {
"expr": xi**2/(2*m) + V0*x + alpha*sp.cos(2*beta),
"dim": 1,
"category": "neutrino",
"description": "Mikheyev–Smirnov–Wolfenstein effect — matter enhancement.",
},
"majorana_mass": {
"expr": xi**2/(2*m) + alpha*x**2 + beta*xi**2,
"dim": 1,
"category": "neutrino",
"description": "Majorana neutrino mass term.",
},
"sterile_neutrino": {
"expr": (xi**2 + eta**2)/(2*m) + eps*(x - y)**2,
"dim": 2,
"category": "neutrino",
"description": "Sterile neutrino mixing — dark sector coupling.",
},
}
# =====================================================================
# 48. MATIÈRE ÉTRANGE & ÉTATS EXOTIQUES
# =====================================================================
H_EXOTIC_MATTER = {
"quark_gluon_plasma": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2)*sp.log(x**2 + y**2 + eps),
"dim": 2,
"category": "exotic_matter",
"description": "Quark–gluon plasma — deconfined QCD matter.",
},
"color_superconductor": {
"expr": (xi**2 + eta**2)/(2*m) - delta*sp.cos(x - y),
"dim": 2,
"category": "exotic_matter",
"description": "Color superconductivity — quark Cooper pairs.",
},
"strangelets": {
"expr": xi**2/(2*m) + alpha*sp.Abs(x)**(4/3),
"dim": 1,
"category": "exotic_matter",
"description": "Strangelet — hypothetical strange quark matter.",
},
"pentaquark": {
"expr": (xi**2 + eta**2)/2 + V0/(sp.Abs(x - y) + eps) + alpha*(x**2 + y**2),
"dim": 2,
"category": "exotic_matter",
"description": "Pentaquark state — exotic hadron.",
},
"tetraquark": {
"expr": (xi**2 + eta**2)/2 - alpha/sp.Abs(x - y) + beta*sp.Abs(x - y),
"dim": 2,
"category": "exotic_matter",
"description": "Tetraquark — four-quark bound state.",
},
}
# =====================================================================
# 49. INFORMATION QUANTIQUE AVANCÉE
# =====================================================================
H_QUANTUM_INFO_ADVANCED = {
"quantum_discord": {
"expr": -(x**2 + y**2)*sp.log(x**2 + y**2 + eps)/2,
"dim": 2,
"category": "quantum_info_advanced",
"description": "Quantum discord — beyond-entanglement correlations.",
},
"measurement_back_action": {
"expr": xi**2/(2*m) + k*x**2/2 + gamma*xi**2*x**2,
"dim": 1,
"category": "quantum_info_advanced",
"description": "Measurement back-action — Heisenberg uncertainty.",
},
"quantum_zeno": {
"expr": xi**2/(2*m) + V0*sp.exp(-alpha*x**2)*(1 - sp.exp(-beta)),
"dim": 1,
"category": "quantum_info_advanced",
"description": "Quantum Zeno effect — inhibition by measurement.",
},
"contextuality": {
"expr": -(sp.cos(x) + sp.cos(y) + sp.cos(x + y)),
"dim": 2,
"category": "quantum_info_advanced",
"description": "Quantum contextuality — Kochen–Specker theorem.",
},
"entanglement_swapping": {
"expr": (xi**2 + eta**2)/2 - alpha*(x*y + xi*eta),
"dim": 2,
"category": "quantum_info_advanced",
"description": "Entanglement swapping protocol.",
},
}
# =====================================================================
# 50. SYSTÈMES PUREMENT MATHÉMATIQUES & ABSTRAITS
# =====================================================================
H_PURE_MATH = {
"riemann_hypothesis": {
"expr": xi**2/(2*m) + sp.re(sp.zeta(0.5 + sp.I*x)),
"dim": 1,
"category": "pure_math",
"description": "Riemann zeta on critical line — analytic number theory.",
},
"modular_form": {
"expr": (xi**2 + eta**2)/(2*(sp.im(x + sp.I*y))**2),
"dim": 2,
"category": "pure_math",
"description": "Modular form geodesic — automorphic functions.",
},
"fibonacci_potential": {
"expr": xi**2/(2*m) + V0*sp.cos(2*sp.pi*x/(1 + sp.sqrt(5))/2),
"dim": 1,
"category": "pure_math",
"description": "Fibonacci quasicrystal — golden ratio modulation.",
},
"cantor_set": {
"expr": xi**2/(2*m) + sum([V0*sp.exp(-alpha*(x - 3**(-n))**2) for n in range(1, 8)]),
"dim": 1,
"category": "pure_math",
"description": "Cantor set potential — fractal energy landscape.",
},
"mandelbrot_escape": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.log(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "pure_math",
"description": "Mandelbrot set escape dynamics.",
},
"julia_set": {
"expr": (xi**2 + eta**2)/2 + V0/(1 + (x**2 + y**2)**2),
"dim": 2,
"category": "pure_math",
"description": "Julia set — complex dynamics.",
},
}
# =====================================================================
# 51. PHÉNOMÉNOLOGIE AU-DELÀ DU MODÈLE STANDARD
# =====================================================================
H_BSM = {
"little_higgs": {
"expr": xi**2/(2*m) + V0*(x**2 + y**2) - alpha*(x**2 + y**2)**2 + beta*(x**4 + y**4),
"dim": 2,
"category": "bsm",
"description": "Little Higgs models — natural EWSB.",
},
"composite_higgs": {
"expr": (xi**2 + eta**2)/2 + V0*sp.sin(sp.sqrt(x**2 + y**2)/alpha)**2,
"dim": 2,
"category": "bsm",
"description": "Composite Higgs — strongly coupled dynamics.",
},
"extra_dimension_kk": {
"expr": (xi**2 + eta**2)/(2*m) + sum([alpha*n**2/(x**2 + y**2 + eps) for n in range(1, 6)]),
"dim": 2,
"category": "bsm",
"description": "Kaluza–Klein tower — extra dimensions.",
},
"technicolor": {
"expr": (xi**2 + eta**2)/2 + V0*sp.cos(x/alpha)*sp.cos(y/beta),
"dim": 2,
"category": "bsm",
"description": "Technicolor — dynamical electroweak symmetry breaking.",
},
"leptoquark": {
"expr": (xi**2 + eta**2)/(2*m) + alpha/(x**2 + y**2 + eps) + beta*(x + y),
"dim": 2,
"category": "bsm",
"description": "Leptoquark interaction — lepton-quark unification.",
},
}
# =====================================================================
# 52. PHYSIQUE DES TROUS NOIRS
# =====================================================================
H_BLACK_HOLES = {
"schwarzschild_particle": {
"expr": (1 - 2*m/sp.sqrt(x**2 + y**2 + eps))*xi**2/2 + eta**2/(2*(1 - 2*m/sp.sqrt(x**2 + y**2 + eps))),
"dim": 2,
"category": "black_holes",
"description": "Particle in Schwarzschild spacetime.",
},
"kerr_geodesic": {
"expr": xi**2/2 + eta**2/(2*(1 + alpha*sp.cos(y)**2)) + beta*(x*eta)/(1 + alpha*sp.cos(y)**2),
"dim": 2,
"category": "black_holes",
"description": "Kerr black hole geodesic — rotating BH.",
},
"hawking_radiation": {
"expr": xi**2/(2*m) - alpha/x + beta*sp.exp(-gamma*x),
"dim": 1,
"category": "black_holes",
"description": "Hawking radiation effective potential.",
},
"ads_black_hole": {
"expr": (1 - m/x**2 - x**2)*xi**2/2 + V0*x**2,
"dim": 1,
"category": "black_holes",
"description": "Anti-de Sitter black hole.",
},
"information_paradox": {
"expr": xi**2/(2*m) + alpha*x**2 - beta*sp.log(x**2 + eps),
"dim": 1,
"category": "black_holes",
"description": "Black hole information paradox model.",
},
}
# =====================================================================
# FIELD THEORY CORRECTED
# =====================================================================
H_FIELD_THEORY_PROPER = {
"klein_gordon_field": {
"expr": (xi**2 + eta**2)/2 + m**2*(x**2 + y**2)/2 + lambda_param*(x**2 + y**2)**2/4,
"dim": 2,
"category": "field_theory",
"description": "Klein–Gordon field — scalar field with self-interaction.",
},
"sine_gordon_field": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - sp.cos(x))*(1 - sp.cos(y)),
"dim": 2,
"category": "field_theory",
"description": "Sine-Gordon field — 2D discretized field theory.",
},
"phi_fourth_field": {
"expr": (xi**2 + eta**2)/2 + m**2*(x**2 + y**2)/2 + lambda_param*(x**4 + y**4)/4,
"dim": 2,
"category": "field_theory",
"description": "φ⁴ theory — quartic self-interaction.",
},
"coupled_oscillator_chain": {
"expr": (xi**2 + eta**2)/(2*m) + k*(x**2 + y**2)/2 + alpha*(x - y)**2/2,
"dim": 2,
"category": "field_theory",
"description": "Coupled oscillator chain — lattice field theory approximation.",
},
}
# =====================================================================
# 53. CONTROL THEORY & OPTIMIZATION
# =====================================================================
H_CONTROL_THEORY = {
"lqr_problem": {
"expr": alpha*x*xi + k*x**2/2 - beta*xi**2/2,
"dim": 1,
"category": "control_theory",
"description": "LQR Problem (Linear Quadratic Regulator) 1D — Pontryagin Hamiltonian after control optimization.",
},
"bang_bang_control": {
"expr": xi*y + alpha*sp.Abs(eta),
"dim": 2,
"category": "control_theory",
"description": "Bang-Bang Control (double integrator) — Pontryagin Hamiltonian for a minimum-time problem with bounded control.",
},
"optimal_braking": {
"expr": xi*y + (eta**2)/(4*alpha) - k*x,
"dim": 2,
"category": "control_theory",
"description": "Optimal Braking Problem (quadratic cost on control) — H = p*v + V(x) + u^2/(2a).",
},
"fuller_problem": {
"expr": xi*y + eta*sp.sign(x),
"dim": 2,
"category": "control_theory",
"description": "Fuller Problem — Double integrator system with a cost $x^2$.",
},
}
# =====================================================================
# 54. ACOUSTICS & WAVE DYNAMICS
# =====================================================================
H_ACOUSTICS = {
"helmholtz_homogeneous": {
"expr": alpha * sp.sqrt(xi**2 + eta**2),
"dim": 2,
"category": "acoustics",
"description": "Helmholtz Equation (homogeneous medium) — Dispersion relation $\omega = c|k|$ (geometrical acoustics).",
},
"acoustic_gradient": {
"expr": (xi**2 + eta**2)/2 - (1 + alpha*x)**2 / 2,
"dim": 2,
"category": "acoustics",
"description": "Acoustic wave in a medium with a refractive index gradient $n(x) = 1 + \alpha x$ — Ray refraction.",
},
"acoustic_waveguide": {
"expr": (xi**2 + eta**2)/2 - V0*sp.exp(-alpha*x**2),
"dim": 2,
"category": "acoustics",
"description": "Acoustic waveguide (SOFAR channel type) — Gaussian index profile creating a potential well.",
},
"paraxial_wave": {
"expr": xi**2 / (2*m) + eta**2 / (2*m) + k*x,
"dim": 2,
"category": "acoustics",
"description": "Paraxial wave equation (Schrödinger analog) — Acoustic beam propagation.",
},
}
# =====================================================================
# 55. COMPLEX NETWORK DYNAMICS
# =====================================================================
H_NETWORK_DYNAMICS = {
"kuramoto_hamiltonian": {
"expr": (xi**2 + eta**2)/(2*m) - k*sp.cos(x - y),
"dim": 2,
"category": "network_dynamics",
"description": "Kuramoto model (N=2) formulated as a Hamiltonian (XY model analog) — Study of synchronization.",
},
"network_consensus": {
"expr": (xi**2 + eta**2)/(2*m) + k*(x - y)**2/2,
"dim": 2,
"category": "network_dynamics",
"description": "Consensus dynamics (N=2) — Diffusion on a graph (quadratic harmonic potential).",
},
"kuramoto_chain_3": {
"expr": (xi**2 + eta**2)/2 + k*(sp.cos(x - y) + sp.cos(y - x)), # Note: just an example, needs 3 vars
"dim": 2, # Note: This is simplified for 2D
"category": "network_dynamics",
"description": "Chain of Kuramoto oscillators (simplified N=3) — Phase interaction potential.",
},
"hopfield_potential": {
"expr": -(xi**2 + eta**2)/2 + alpha*(x**2 - 1)**2 + beta*(y**2 - 1)**2 - k*x*y,
"dim": 2,
"category": "network_dynamics",
"description": "Hopfield network potential (N=2) — Dynamics of an associative memory.",
},
}
# =====================================================================
# 56. SPIN GLASSES & DISORDERED SYSTEMS
# =====================================================================
H_SPIN_GLASS = {
"sk_model_potential": {
"expr": -alpha*(x**2 + y**2) + beta*(x**4 + y**4) + gamma*(x*y)**2,
"dim": 2,
"category": "spin_glass",
"description": "Sherrington-Kirkpatrick (SK) potential — Free energy (2-spin replicas) showing a complex energy landscape.",
},
"edwards_anderson_pheno": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x) + sp.cos(y) + alpha*sp.cos(x - y + beta)),
"dim": 2,
"category": "spin_glass",
"description": "Edwards-Anderson model (phenomenological) — Frustrated and disordered periodic potential.",
},
"p_spin_potential": {
"expr": -alpha*(x**3 + y**3) - beta*x*y,
"dim": 2,
"category": "spin_glass",
"description": "'p-spin' model (p=3, N=2) — Complex energy landscape, model for the glass transition.",
},
"random_field_ising": {
"expr": xi**2/(2*m) - k*sp.cos(x) - alpha*x,
"dim": 1,
"category": "spin_glass",
"description": "Random Field Ising Model (RFIM) — Ferromagnetic interaction + disordered field (here constant).",
},
}
# =====================================================================
# 57. MESOSCOPIC PHYSICS
# =====================================================================
H_MESOSCOPIC = {
"coulomb_blockade": {
"expr": alpha*(x - beta)**2 + k*xi**2,
"dim": 1,
"category": "mesoscopic",
"description": "Coulomb blockade (Quantum Dot) — Charging energy $E_C(N-N_g)^2$ (x=Charge, xi=Phase).",
},
"caldeira_leggett": {
"expr": xi**2/(2*m) + alpha*x**2/2 + eta**2/2 + k*y**2/2 + gamma*x*y,
"dim": 2,
"category": "mesoscopic",
"description": "Caldeira-Leggett model (1 mode) — Decoherence of a quantum oscillator (x) coupled to a bath (y).",
},
"luttinger_liquid_2mode": {
"expr": (xi**2 + eta**2)/2 + k*(x**2 + y**2)/2 + alpha*(x - y)**2,
"dim": 2,
"category": "mesoscopic",
"description": "Luttinger liquid (2-mode approximation) — Bosonization of 1D fermions (plasmon/spin modes).",
},
"aharonov_bohm_ring": {
"expr": (xi - alpha)**2/(2*m) + k*x**2,
"dim": 1,
"category": "mesoscopic",
"description": "Aharonov-Bohm ring (1D) — Oscillator with magnetic flux $\\alpha$ shifting the canonical momentum.",
},
}
# =====================================================================
# 58. POLYMER PHYSICS
# =====================================================================
H_POLYMERS = {
"edwards_polymer": {
"expr": xi**2/(2*m) + k*x**2/2 + alpha*x**4,
"dim": 1,
"category": "polymers",
"description": "Edwards model (φ⁴ field theory) — Polymer chain with excluded volume interaction.",
},
"flory_huggins_energy": {
"expr": k*(x*sp.log(x + eps) + (1-x)*sp.log(1-x + eps)) + alpha*x*(1-x),
"dim": 1,
"category": "polymers",
"description": "Flory-Huggins free energy — Polymer mixture theory (x = volume fraction).",
},
"fjc_potential": {
"expr": xi**2/(2*m) - alpha*sp.log(sp.sinh(k*x + eps)/(k*x + eps)),
"dim": 1,
"category": "polymers",
"description": "Freely Jointed Chain (FJC) — Effective potential (Langevin approximation) for stretching.",
},
"worm_like_chain": {
"expr": xi**2/(2*m) + k*x**2/(2*(1-x/alpha)),
"dim": 1,
"category": "polymers",
"description": "Worm-like Chain (WLC) model — Elasticity for a semi-flexible polymer (x=extension, α=max length).",
},
}
# =====================================================================
# 59. TOPOLOGICAL FIELD THEORIES (TFT)
# =====================================================================
H_TFT = {
"chern_simons_abelian": {
"expr": (x*eta - y*xi),
"dim": 2,
"category": "tft",
"description": "Abelian Chern–Simons term — topological action for quantum Hall effect.",
},
"bf_model": {
"expr": x*eta - y*xi + V0*(x**2 + y**2),
"dim": 2,
"category": "tft",
"description": "BF theory in 2+1D (simplified phase-space form) — topological gravity analog.",
},
"wdvv_potential": {
"expr": xi**2/(2*m) + sp.log(sp.Abs(sp.diff(V0*sp.exp(-x**2), x, 3))),
"dim": 1,
"category": "tft",
"description": "WDVV equation from Frobenius manifold — enumerative geometry link.",
},
"achiral_tqft": {
"expr": (xi**2 + eta**2)/2 + alpha*(x*eta - y*xi)**2,
"dim": 2,
"category": "tft",
"description": "Effective Hamiltonian for achiral topological quantum field theory.",
},
}
# =====================================================================
# 60. ADVANCED GEOMETRIC HAMILTONIANS
# =====================================================================
H_GEOMETRIC_ADVANCED = {
"sphere_geodesic": {
"expr": (xi**2 + eta**2)/(2*(sp.cos(y)**2 + eps)),
"dim": 2,
"category": "geometric_advanced",
"description": "Geodesic flow on a sphere (latitude/longitude coordinates).",
},
"torus_geodesic": {
"expr": (xi**2 + eta**2)/(2*(1 + 0.5*sp.cos(y))),
"dim": 2,
"category": "geometric_advanced",
"description": "Geodesic on a torus with major/minor radii ratio = 2.",
},
"ellipsoid_geodesic": {
"expr": (xi**2 + eta**2)/(2*(1 + alpha*sp.cos(x)**2 + beta*sp.sin(y)**2)),
"dim": 2,
"category": "geometric_advanced",
"description": "Geodesic on ellipsoid — non-constant curvature.",
},
"variable_curvature": {
"expr": (xi**2 + eta**2)/(2*(1 + alpha*sp.cos(x)**2)),
"dim": 2,
"category": "geometric_advanced",
"description": "2D metric with spatially varying Gaussian curvature.",
},
"neumann_oscillator": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2 + beta*y**2,
"dim": 2,
"category": "geometric_advanced",
"description": "Neumann system — particle on sphere with quadratic potential.",
},
"calogero_sutherland": {
"expr": (xi**2 + eta**2)/2 + g/(sp.sin((x - y)/2)**2 + eps),
"dim": 2,
"category": "geometric_advanced",
"description": "Calogero–Sutherland model — integrable with trigonometric interaction.",
},
"birkhoff_normal_form": {
"expr": xi**2/2 + eta**2/2 + alpha*(x**2 + y**2)**2 + beta*(x**4 - y**4),
"dim": 2,
"category": "geometric_advanced",
"description": "4th-order Birkhoff normal form near elliptic equilibrium.",
},
}
# =====================================================================
# 61. SYMMETRIES AND HAMILTONIAN REDUCTIONS
# =====================================================================
H_SYMMETRY_REDUCED = {
"particle_on_sphere": {
"expr": (xi**2 + eta**2)/(2*m) + lambda_param*(x**2 + y**2 - R**2),
"dim": 2,
"category": "symmetry_reduced",
"description": "Particle constrained to sphere via Lagrange multiplier (Dirac formalism).",
},
"reduced_rotor": {
"expr": xi**2/(2*sp.I) + omega*L_z,
"dim": 1,
"category": "symmetry_reduced",
"description": "Symplectic reduction of rotating rigid body (L_z = const).",
},
"gauge_invariant_oscillator": {
"expr": ((xi - A*y)**2 + (eta + A*x)**2)/(2*m),
"dim": 2,
"category": "symmetry_reduced",
"description": "U(1)-gauge invariant oscillator — conserved angular momentum.",
},
"magnetic_monopole_reduced": {
"expr": (xi**2 + eta**2)/(2*m) + g*sp.acos(y/sp.sqrt(x**2 + y**2 + eps)),
"dim": 2,
"category": "symmetry_reduced",
"description": "Dirac monopole with azimuthal symmetry reduction.",
},
}
# =====================================================================
# 62. EXTENDED QUANTUM TOPOLOGICAL & RELATIVISTIC HAMILTONIANS
# =====================================================================
H_QUANTUM_TOPOLOGICAL_EXTENDED = {
"dirac_2d_nonuniform_B": {
"expr": sp.sqrt((xi - A*sp.exp(-x**2)*y)**2 + (eta + A*x*sp.exp(-y**2))**2 + m**2),
"dim": 2,
"category": "quantum_topological_extended",
"description": "2D Dirac with Gaussian magnetic field — Landau levels + edge states.",
},
"weyl_semimetal": {
"expr": sp.sqrt(xi**2 + eta**2) + alpha*(x*eta - y*xi),
"dim": 2,
"category": "quantum_topological_extended",
"description": "Weyl semimetal Hamiltonian — linear dispersion + spin-momentum locking.",
},
"graphene_dirac": {
"expr": xi*sp.cos(2*sp.pi/3) + eta*sp.sin(2*sp.pi/3) + V0*(sp.cos(x) + sp.cos(y)),
"dim": 2,
"category": "quantum_topological_extended",
"description": "Continuum graphene Dirac Hamiltonian near K point.",
},
"majorana_wire": {
"expr": sp.sqrt(xi**2 + Delta**2*sp.sin(x)**2) - mu,
"dim": 1,
"category": "quantum_topological_extended",
"description": "Kitaev chain effective Hamiltonian for Majorana modes.",
},
}
# =====================================================================
# 63. CONTINUUM FIELDS & MULTI-D SOLITONS
# =====================================================================
H_CONTINUUM_SOLITONS = {
"nls_2d_radial": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2)**2,
"dim": 2,
"category": "continuum_solitons",
"description": "2D cubic NLS — collapsing/vortex solitons.",
},
"sine_gordon_2d": {
"expr": (xi**2 + eta**2)/2 + V0*(1 - sp.cos(sp.sqrt(x**2 + y**2))),
"dim": 2,
"category": "continuum_solitons",
"description": "Radially symmetric 2D sine-Gordon — breather analogs.",
},
"kp_ii_equation": {
"expr": (xi**2 + eta**2)/2 + alpha*x**3 + beta*x*eta**2,
"dim": 2,
"category": "continuum_solitons",
"description": "Kadomtsev–Petviashvili II — weakly 2D KdV.",
},
"benjamin_ono": {
"expr": xi**2/2 + alpha*xi*sp.Abs(xi) + beta*x**2,
"dim": 1,
"category": "continuum_solitons",
"description": "Benjamin–Ono equation — Hilbert transform dispersion.",
},
"camassa_holm": {
"expr": xi**2/2 + alpha*x*xi**2 + beta*x**3,
"dim": 1,
"category": "continuum_solitons",
"description": "Camassa–Holm — peakon solutions.",
},
}
# =====================================================================
# 64. ADVANCED STOCHASTIC & DISSIPATIVE SYSTEMS
# =====================================================================
H_STOCHASTIC_ADVANCED = {
"multiplicative_noise": {
"expr": xi**2/2 + alpha*x**2 + beta*x*xi + gamma*xi**2*x,
"dim": 1,
"category": "stochastic_advanced",
"description": "Hamiltonian with multiplicative noise coupling.",
},
"memory_kernel_effective": {
"expr": xi**2/(2*m) + alpha*x**2/2 + beta*sp.exp(-gamma*sp.Abs(xi)),
"dim": 1,
"category": "stochastic_advanced",
"description": "Effective Hamiltonian with memory (non-Markovian bath).",
},
"fokker_planck_nonquadratic": {
"expr": sp.Abs(xi)**3/3 + V0*x**4,
"dim": 1,
"category": "stochastic_advanced",
"description": "Non-quadratic kinetic term from anomalous diffusion.",
},
}
# =====================================================================
# 65. MULTI-SCALE HYBRIDS & CHAOS
# =====================================================================
H_MULTI_SCALE_CHAOS = {
"kam_perturbation": {
"expr": xi**2/2 + eta**2/2 + eps*sp.cos(x)*sp.cos(y),
"dim": 2,
"category": "multi_scale_chaos",
"description": "Near-integrable KAM system — weakly perturbed torus.",
},
# "chirikov_continuous": {
# "expr": (xi**2 + eta**2)/2 + alpha*sp.cos(x)*sp.cos(omega*t),
# "dim": 2,
# "category": "multi_scale_chaos",
# "description": "Continuous analog of Chirikov standard map.",
# },
# "two_timescale": {
# "expr": xi**2/2 + V0*sp.cos(x)*sp.cos(omega*t),
# "dim": 1,
# "category": "multi_scale_chaos",
# "description": "Explicitly time-dependent Hamiltonian (t treated as parameter).",
# },
"quasiperiodic_coupling": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.cos(x + golden_ratio*y),
"dim": 2,
"category": "multi_scale_chaos",
"description": "Incommensurate coupling — Aubry transition precursor.",
},
}
# =====================================================================
# 66. MODERN & COMPUTATIONAL EXTENSIONS
# =====================================================================
H_MODERN_EXTENSIONS = {
"pt_symmetric_2d": {
"expr": xi**2 + eta**2 + sp.I*(x**3 + y**3),
"dim": 2,
"category": "modern_extensions",
"description": "2D PT-symmetric non-Hermitian Hamiltonian.",
},
"quantum_adiabatic": {
"expr": A_param*(xi**2 + x**2) + B_param*(eta**2 + y**2) + C_param*x*y,
"dim": 2,
"category": "modern_extensions",
"description": "Interpolated Hamiltonian for adiabatic quantum computing.",
},
"learned_hamiltonian": {
"expr": xi**2/2 + alpha*sp.tanh(beta*x) + gamma*sp.sin(delta*x),
"dim": 1,
"category": "modern_extensions",
"description": "Symbolic surrogate from Hamiltonian neural ODE.",
},
}
# =====================================================================
# 67. GAME THEORY & EVOLUTIONARY DYNAMICS
# =====================================================================
H_GAME_DYNAMICS = {
"replicator_hamiltonian": {
"expr": xi**2/2 + alpha*x*(1 - x)*(x - beta),
"dim": 1,
"category": "game_dynamics",
"description": "Hamiltonian form of replicator dynamics in 2-strategy evolutionary game.",
},
"hawk_dove_game": {
"expr": xi**2/2 + V0*x*(1 - x) - C_param*x**2/2,
"dim": 1,
"category": "game_dynamics",
"description": "Hawk-Dove game payoff encoded as effective potential.",
},
"prisoner_dilemma_potential": {
"expr": (xi**2 + eta**2)/2 - alpha*(x*y + (1 - x)*(1 - y)),
"dim": 2,
"category": "game_dynamics",
"description": "Potential encoding mutual cooperation vs betrayal in Prisoner's Dilemma.",
},
}
# =====================================================================
# 68. OPTIMIZATION & MACHINE LEARNING
# =====================================================================
H_OPTIMIZATION = {
"nesterov_ode": {
"expr": xi**2/2 + k*x**2/2 + gamma*x*xi,
"dim": 1,
"category": "optimization",
"description": "Continuous-time limit of Nesterov's accelerated gradient descent.",
},
"symplectic_sgd": {
"expr": xi**2/2 + V0*sp.tanh(x)**2,
"dim": 1,
"category": "optimization",
"description": "Symplectic stochastic gradient flow for nonconvex optimization.",
},
"primal_dual_hamiltonian": {
"expr": xi*y - f_param*x - g_param*y,
"dim": 2,
"category": "optimization",
"description": "Hamiltonian formulation of primal-dual optimization (f, g convex).",
},
}
# =====================================================================
# 69. QUANTITATIVE FINANCE
# =====================================================================
H_QUANT_FINANCE = {
"martingale_hamiltonian": {
"expr": xi**2/2 - alpha*sp.log(x + eps),
"dim": 1,
"category": "quant_finance",
"description": "Martingale constraint encoded as Hamiltonian potential (geometric Brownian motion).",
},
"portfolio_optimization": {
"expr": (xi**2 + eta**2)/2 - alpha*x - beta*y + gamma*(x - y)**2,
"dim": 2,
"category": "quant_finance",
"description": "Mean-variance portfolio selection as Hamiltonian system.",
},
"risk_measure_flow": {
"expr": xi**2/2 + V0*sp.exp(-x**2) + alpha*x**4,
"dim": 1,
"category": "quant_finance",
"description": "Dynamic risk measure (e.g., entropic risk) as potential.",
},
}
# =====================================================================
# 70. SYMBOLIC COMPUTATION & REVERSIBLE LOGIC
# =====================================================================
H_SYMBOLIC_COMPUTATION = {
"reversible_automaton": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.Mod(x + y, 2),
"dim": 2,
"category": "symbolic",
"description": "Hamiltonian encoding of a reversible cellular automaton rule.",
},
"logical_gate_potential": {
"expr": xi**2/2 + V0*(x - sp.Piecewise((0, x < 0.5), (1, True)))**2,
"dim": 1,
"category": "symbolic",
"description": "Energy landscape enforcing binary logic behavior (e.g., step function).",
},
}
# =====================================================================
# 71. GENERATIVE DESIGN & MORPHOGENESIS
# =====================================================================
H_GENERATIVE_DESIGN = {
"growth_potential": {
"expr": xi**2/2 + alpha*sp.exp(-x**2) * sp.cos(beta*x),
"dim": 1,
"category": "generative",
"description": "Morphogenetic potential for procedural branching in design.",
},
"turing_pattern_design": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2*y - beta*x,
"dim": 2,
"category": "generative",
"description": "Hamiltonian derived from Turing reaction-diffusion for generative art.",
},
}
# =====================================================================
# 72. EPIDEMIOLOGY & POPULATION DYNAMICS
# =====================================================================
H_EPIDEMIOLOGY = {
"sir_hamiltonian": {
"expr": (xi**2 + eta**2)/2 + alpha*x*y - beta*y,
"dim": 2,
"category": "epidemiology",
"description": "Hamiltonian form of SIR model — susceptible-infected-recovered flow.",
},
"seir_potential": {
"expr": (xi**2 + eta**2 + zeta**2)/2 + alpha*x*y - beta*y - gamma*z,
"dim": 2, # Note: zeta, z not in global vars → simplified to 2D
"category": "epidemiology",
"description": "Reduced SEIR dynamics as effective 2D Hamiltonian (latent variable integrated).",
},
"epidemic_wave": {
"expr": xi**2/2 + alpha*sp.exp(-x**2)*y,
"dim": 2,
"category": "epidemiology",
"description": "Traveling epidemic wave — spatial spread with Gaussian kernel.",
},
"vaccination_game": {
"expr": xi**2/2 + V0*x*(1 - x) - C_param*x,
"dim": 1,
"category": "epidemiology",
"description": "Vaccination decision dynamics — cost-benefit in epidemic risk.",
},
}
# =====================================================================
# 73. LINGUISTICS & SEMIOTIC SYSTEMS
# =====================================================================
H_LINGUISTICS = {
"language_drift": {
"expr": xi**2/2 + alpha*sp.cos(beta*x),
"dim": 1,
"category": "linguistics",
"description": "Phonemic drift as particle in periodic potential — language evolution.",
},
"grammar_potential": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - sp.cos(x - y)),
"dim": 2,
"category": "linguistics",
"description": "Syntactic alignment — energy landscape for grammatical agreement.",
},
"word_embedding_flow": {
"expr": (xi**2 + eta**2)/2 + V0*sp.exp(-alpha*(x**2 + y**2)),
"dim": 2,
"category": "linguistics",
"description": "Effective Hamiltonian for word embedding dynamics in semantic space.",
},
"zipf_law_potential": {
"expr": xi**2/2 - alpha*sp.log(x + eps),
"dim": 1,
"category": "linguistics",
"description": "Zipf’s law as logarithmic potential — frequency vs rank in language.",
},
}
# =====================================================================
# 74. ECOLOGY & ECOSYSTEM NETWORKS
# =====================================================================
H_ECOLOGY = {
"lotka_volterra_hamiltonian": {
"expr": (xi**2 + eta**2)/2 + alpha*x*y - beta*x - gamma*y,
"dim": 2,
"category": "ecology",
"description": "Hamiltonian formulation of Lotka–Volterra predator-prey dynamics.",
},
"competitive_exclusion": {
"expr": (xi**2 + eta**2)/2 - alpha*x**2 - beta*y**2 + gamma*x*y,
"dim": 2,
"category": "ecology",
"description": "Competition model — niche overlap and exclusion principle.",
},
"mutualism_potential": {
"expr": xi**2/2 + eta**2/2 - alpha*sp.log(1 + x) - beta*sp.log(1 + y) + gamma*x*y,
"dim": 2,
"category": "ecology",
"description": "Mutualistic interaction — cooperative species benefit.",
},
"trophic_cascade": {
"expr": (xi**2 + eta**2)/2 + alpha*x - beta*x*y + gamma*y*z, # z not defined → use 2D proxy
"dim": 2,
"category": "ecology",
"description": "Simplified trophic cascade (top-down control) as 2D effective Hamiltonian.",
},
}
# =====================================================================
# 75. MACHINE LEARNING & PROBABILISTIC INFERENCE
# =====================================================================
H_INFERENCE = {
"variational_free_energy": {
"expr": xi**2/2 + alpha*x**2/2 + beta*sp.log(sp.cosh(x)),
"dim": 1,
"category": "inference",
"description": "Variational free energy — inference as energy minimization.",
},
"expectation_propagation": {
"expr": (xi**2 + eta**2)/2 + 0.5*(x - mu1)**2/sigma1**2 + 0.5*(y - mu2)**2/sigma2**2 - alpha*x*y,
"dim": 2,
"category": "inference",
"description": "Expectation propagation — Gaussian message passing as coupled oscillators.",
},
"information_geometry": {
"expr": (xi**2 + eta**2)/(2*(1 + alpha*x**2)),
"dim": 2,
"category": "inference",
"description": "Fisher–Rao metric as curved Hamiltonian phase space.",
},
"diffusion_inference": {
"expr": sp.Abs(xi)**alpha + V0*x**2/2,
"dim": 1,
"category": "inference",
"description": "Score-based diffusion models — Lévy-driven inference dynamics.",
},
}
# =====================================================================
# 76. URBAN DYNAMICS
# =====================================================================
H_URBAN = {
"traffic_flow": {
"expr": xi**2/2 + V0*x*(1 - x),
"dim": 1,
"category": "urban",
"description": "Lighthill–Whitham traffic model — density-dependent flow potential.",
},
"land_use_competition": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2*(1 - x) - beta*y**2 + gamma*x*y,
"dim": 2,
"category": "urban",
"description": "Competition between residential (x) and commercial (y) land use.",
},
"urban_heat_island": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.exp(-beta*(x**2 + y**2)),
"dim": 2,
"category": "urban",
"description": "Urban heat island effect — temperature gradient as potential well.",
},
"pedestrian_evacuation": {
"expr": xi**2/(2*m) - alpha*sp.log(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "urban",
"description": "Pedestrian escape dynamics — logarithmic attraction to exit.",
},
}
# =====================================================================
# 77. COGNITIVE SCIENCE
# =====================================================================
H_COGNITIVE = {
"belief_updating": {
"expr": xi**2/2 + alpha*(x - beta)**2/2 - gamma*sp.log(sp.cosh(x)),
"dim": 1,
"category": "cognitive",
"description": "Bayesian belief updating — Gaussian prior with logistic evidence.",
},
"attention_potential": {
"expr": (xi**2 + eta**2)/2 + V0*sp.exp(-alpha*(x**2 + y**2)),
"dim": 2,
"category": "cognitive",
"description": "Spatial attention field — Gaussian focus of perceptual resources.",
},
"predictive_coding": {
"expr": xi**2/2 + alpha*(x - y)**2/2,
"dim": 2,
"category": "cognitive",
"description": "Predictive coding error — minimization of prediction vs sensation.",
},
"working_memory": {
"expr": xi**2/2 + alpha*x**4 - beta*x**2,
"dim": 1,
"category": "cognitive",
"description": "Bistable working memory — winner-take-all attractor dynamics.",
},
}
# =====================================================================
# 78. LEGAL SYSTEMS
# =====================================================================
H_LEGAL = {
"norm_diffusion": {
"expr": xi**2/(2*m) - alpha*sp.log(1 + sp.Abs(x)),
"dim": 1,
"category": "legal",
"description": "Diffusion of legal norms — logarithmic penalty for deviation.",
},
"precedent_flow": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.tanh(x)*sp.tanh(y),
"dim": 2,
"category": "legal",
"description": "Precedent alignment — mutual reinforcement of case outcomes.",
},
"legal_entropy": {
"expr": -x*sp.log(x + eps) - (1 - x)*sp.log(1 - x + eps) + alpha*x*xi,
"dim": 1,
"category": "legal",
"description": "Legal uncertainty (entropy) as potential — binary legal states.",
},
"jurisprudential_tension": {
"expr": xi**2/2 + alpha*sp.cos(beta*x),
"dim": 1,
"category": "legal",
"description": "Cyclic legal interpretation — oscillation between doctrines.",
},
}
# =====================================================================
# 79. ART & MUSIC
# =====================================================================
H_ART_MUSIC = {
"harmonic_tension": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/12)),
"dim": 1,
"category": "art_music",
"description": "Harmonic tension in 12-tone equal temperament — circular pitch space.",
},
"consonance_potential": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - sp.cos(x - y)),
"dim": 2,
"category": "art_music",
"description": "Consonance model — energy minimized at simple frequency ratios.",
},
"generative_composition": {
"expr": (xi**2 + eta**2)/2 + V0*sp.sin(alpha*x)*sp.sin(beta*y),
"dim": 2,
"category": "art_music",
"description": "Lissajous-inspired generative music — beat and phasing dynamics.",
},
"color_harmony": {
"expr": (xi**2 + eta**2)/2 + V0*(1 - sp.cos(sp.atan2(y, x))),
"dim": 2,
"category": "art_music",
"description": "Color harmony on hue circle — angular similarity in HSV space.",
},
}
# =====================================================================
# 80. EDUCATION & LEARNING DYNAMICS
# =====================================================================
H_EDUCATION = {
"learning_curve": {
"expr": xi**2/2 + V0*(1 - sp.exp(-alpha*x)),
"dim": 1,
"category": "education",
"description": "Learning curve — diminishing returns in skill acquisition.",
},
"knowledge_diffusion": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.log(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "education",
"description": "Diffusion of knowledge — logarithmic attraction in idea space.",
},
"forgetting_potential": {
"expr": xi**2/2 + alpha*x**2*sp.exp(-beta*x),
"dim": 1,
"category": "education",
"description": "Ebbinghaus forgetting curve — memory decay with rehearsal.",
},
"curriculum_design": {
"expr": (xi**2 + eta**2)/2 + alpha*(x - y)**2 + beta*x**2,
"dim": 2,
"category": "education",
"description": "Curriculum scaffolding — alignment of prior and new knowledge.",
},
}
# =====================================================================
# 81. RELIGION & DOCTRINAL DYNAMICS
# =====================================================================
H_RELIGION = {
"doctrinal_evolution": {
"expr": xi**2/2 + V0*sp.cos(alpha*x),
"dim": 1,
"category": "religion",
"description": "Doctrinal oscillation — cyclic reinterpretation of sacred texts.",
},
"ritual_periodicity": {
"expr": xi**2/2 + alpha*(1 - sp.cos(2*sp.pi*x/T)),
"dim": 1,
"category": "religion",
"description": "Ritual timing — harmonic potential for liturgical cycles.",
},
"sectarian_splitting": {
"expr": (xi**2 + eta**2)/2 - alpha*x**2 - beta*y**2 + gamma*(x - y)**4,
"dim": 2,
"category": "religion",
"description": "Sect formation — symmetry breaking in belief space.",
},
"religious_entropy": {
"expr": -x*sp.log(x + eps) - (1 - x)*sp.log(1 - x + eps) + alpha*x*xi,
"dim": 1,
"category": "religion",
"description": "Uncertainty in belief commitment — binary doctrinal states.",
},
}
# =====================================================================
# 82. SPORTS & TACTICAL DYNAMICS
# =====================================================================
H_SPORTS = {
"tactical_flow": {
"expr": (xi**2 + eta**2)/2 + V0*sp.exp(-alpha*(x**2 + y**2)),
"dim": 2,
"category": "sports",
"description": "Team tactical focus — Gaussian attractor in field space.",
},
"player_interaction": {
"expr": (xi**2 + eta**2)/2 + alpha/(sp.sqrt((x - y)**2 + eps)) - beta*(x + y)**2,
"dim": 2,
"category": "sports",
"description": "Player coupling — attraction/repulsion in cooperative play.",
},
"game_momentum": {
"expr": xi**2/2 + alpha*sp.tanh(beta*x)*x,
"dim": 1,
"category": "sports",
"description": "Psychological momentum — nonlinear reinforcement of success.",
},
"zone_defense": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.Abs(x) + beta*sp.Abs(y),
"dim": 2,
"category": "sports",
"description": "Zone defense potential — piecewise linear spatial control.",
},
}
# =====================================================================
# 83. AGRICULTURE & ECOLOGICAL MANAGEMENT
# =====================================================================
H_AGRICULTURE = {
"crop_rotation": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.cos(2*sp.pi*x/3) + beta*sp.cos(2*sp.pi*y/3),
"dim": 2,
"category": "agriculture",
"description": "Three-field crop rotation — periodic soil nutrient cycling.",
},
"pest_predator": {
"expr": (xi**2 + eta**2)/2 + alpha*x - beta*x*y + gamma*y,
"dim": 2,
"category": "agriculture",
"description": "Pest dynamics with biological control — Lotka–Volterra analog.",
},
"soil_nutrient_diffusion": {
"expr": xi**2/(2*m) - alpha*sp.log(x + eps) + beta*x,
"dim": 1,
"category": "agriculture",
"description": "Soil fertility gradient — logarithmic depletion + replenishment.",
},
"drought_response": {
"expr": xi**2/2 + V0*sp.exp(-alpha*x**2) + beta*x**4,
"dim": 1,
"category": "agriculture",
"description": "Plant stress response — resilience under water scarcity.",
},
}
# =====================================================================
# 84. PUBLIC HEALTH
# =====================================================================
H_PUBLIC_HEALTH = {
"vaccination_campaign": {
"expr": xi**2/2 + V0*x*(1 - x) - alpha*x,
"dim": 1,
"category": "public_health",
"description": "Vaccination uptake dynamics — logistic coverage with cost penalty.",
},
"epidemic_preparedness": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.log(1 + x**2 + y**2),
"dim": 2,
"category": "public_health",
"description": "Preparedness as attraction to central response hub — logarithmic potential.",
},
"herd_immunity_threshold": {
"expr": xi**2/2 - alpha*(x - beta)**2/2,
"dim": 1,
"category": "public_health",
"description": "Herd immunity as stable equilibrium — Gaussian well at critical coverage.",
},
"contact_tracing_flow": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.exp(-beta*(x**2 + y**2)),
"dim": 2,
"category": "public_health",
"description": "Contact tracing as localized information potential — Gaussian kernel.",
},
}
# =====================================================================
# 85. ARCHITECTURE
# =====================================================================
H_ARCHITECTURE = {
"structural_flow": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2)**2,
"dim": 2,
"category": "architecture",
"description": "Structural load distribution — quartic stiffness in planar frame.",
},
"spatial_perception": {
"expr": (xi**2 + eta**2)/2 - V0*sp.log(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "architecture",
"description": "Perceptual attraction to center — logarithmic spatial focus.",
},
"circulation_potential": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.Abs(x) + beta*sp.Abs(y),
"dim": 2,
"category": "architecture",
"description": "Pedestrian circulation — piecewise linear corridor constraints.",
},
"proportion_harmony": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/golden_ratio)),
"dim": 1,
"category": "architecture",
"description": "Aesthetic proportion — harmonic potential at golden ratio intervals.",
},
}
# =====================================================================
# 86. CUISINE
# =====================================================================
H_CUISINE = {
"flavor_pairing": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.exp(-beta*(x - y)**2),
"dim": 2,
"category": "cuisine",
"description": "Flavor compatibility — Gaussian attraction between similar tastes.",
},
"umami_potential": {
"expr": xi**2/2 + V0/(1 + sp.exp(-alpha*x)) - beta*x**2,
"dim": 1,
"category": "cuisine",
"description": "Umami taste response — sigmoid activation with saturation.",
},
"recipe_dynamics": {
"expr": (xi**2 + eta**2)/2 + alpha*x*y*(1 - x - y),
"dim": 2,
"category": "cuisine",
"description": "Recipe balance — ternary constraint (x + y ≤ 1) for ingredient ratios.",
},
"cooking_time_opt": {
"expr": xi**2/(2*m) + alpha*(x - beta)**2 + gamma*sp.exp(-delta*x),
"dim": 1,
"category": "cuisine",
"description": "Optimal cooking time — trade-off between doneness and degradation.",
},
}
# =====================================================================
# 87. FASHION
# =====================================================================
H_FASHION = {
"trend_diffusion": {
"expr": xi**2/(2*m) - alpha*sp.log(1 + sp.Abs(x)),
"dim": 1,
"category": "fashion",
"description": "Trend adoption — logarithmic resistance to deviation from norm.",
},
"style_cycles": {
"expr": xi**2/2 + V0*sp.cos(alpha*x),
"dim": 1,
"category": "fashion",
"description": "Cyclic revival of styles — periodic potential over decades.",
},
"aesthetic_tension": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - sp.cos(x - y)),
"dim": 2,
"category": "fashion",
"description": "Outfit coherence — consonance between garment elements.",
},
"fast_fashion_dissipation": {
"expr": xi**2/2 + beta*x**2 - gamma*x**3,
"dim": 1,
"category": "fashion",
"description": "Fast fashion decay — rapid trend obsolescence (cubic instability).",
},
}
# =====================================================================
# 88. RELAXATION & WELLNESS
# =====================================================================
H_WELLNESS = {
"stress_recovery": {
"expr": xi**2/2 + alpha*x**2*sp.exp(-beta*x),
"dim": 1,
"category": "wellness",
"description": "Stress decay with nonlinear recovery — psychological resilience model.",
},
"circadian_rhythm": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/24)),
"dim": 1,
"category": "wellness",
"description": "Circadian cycle — 24-hour periodic biological oscillator.",
},
"heart_rate_variability": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - sp.cos(x - y)),
"dim": 2,
"category": "wellness",
"description": "HRV coherence — coupling between respiration and heart rate.",
},
"meditation_potential": {
"expr": xi**2/2 - alpha*sp.log(sp.cosh(x)) + beta*x**2,
"dim": 1,
"category": "wellness",
"description": "Meditative state as energy well — entropy reduction in mental noise.",
},
}
# =====================================================================
# 89. DIGITAL CULTURE
# =====================================================================
H_DIGITAL_CULTURE = {
"meme_spread": {
"expr": xi**2/(2*m) - alpha*sp.log(1 + sp.Abs(x)) + beta*x**2,
"dim": 1,
"category": "digital_culture",
"description": "Meme virality — logarithmic resistance to novelty saturation.",
},
"algorithmic_bias": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.tanh(x)*sp.tanh(y),
"dim": 2,
"category": "digital_culture",
"description": "Feedback loop in recommendation systems — polarization attractor.",
},
"attention_economy": {
"expr": xi**2/2 - V0*sp.exp(-alpha*x**2),
"dim": 1,
"category": "digital_culture",
"description": "Attention as scarce resource — Gaussian capture by content.",
},
"digital_echo_chamber": {
"expr": (xi**2 + eta**2)/2 - alpha*x*y + beta*(x**2 + y**2),
"dim": 2,
"category": "digital_culture",
"description": "Echo chamber formation — alignment reinforced by platform design.",
},
}
# =====================================================================
# 90. URBAN MYTH & FOLKLORE
# =====================================================================
H_FOLKLORE = {
"narrative_diffusion": {
"expr": xi**2/(2*m) - alpha*sp.log(sp.sqrt(x**2 + eps)),
"dim": 1,
"category": "folklore",
"description": "Myth propagation — inverse-square attenuation with distance.",
},
"rumor_dynamics": {
"expr": xi**2/2 + V0*x*(1 - x)*(x - beta),
"dim": 1,
"category": "folklore",
"description": "Rumor spread — bistable potential between belief and skepticism.",
},
"archetype_potential": {
"expr": xi**2/2 + alpha*(1 - sp.cos(2*sp.pi*x)),
"dim": 1,
"category": "folklore",
"description": "Jungian archetype cycle — periodic recurrence in cultural narratives.",
},
"legend_persistence": {
"expr": (xi**2 + eta**2)/2 + V0*sp.exp(-alpha*(x**2 + y**2)),
"dim": 2,
"category": "folklore",
"description": "Urban legend as localized attractor — spatial-temporal memory kernel.",
},
}
# =====================================================================
# 91. PERFUMERY
# =====================================================================
H_PERFUMERY = {
"scent_harmony": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - sp.cos(x - y)),
"dim": 2,
"category": "perfumery",
"description": "Olfactory consonance — energy minimized at balanced note ratios.",
},
"volatility_gradient": {
"expr": xi**2/2 + alpha*sp.exp(-beta*x),
"dim": 1,
"category": "perfumery",
"description": "Top-middle-base note volatility — exponential decay of scent intensity.",
},
"fragrance_accord": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.exp(-alpha*x**2) + sp.exp(-beta*y**2) + gamma*sp.exp(-delta*(x-y)**2)),
"dim": 2,
"category": "perfumery",
"description": "Perfume accord — blend of Gaussian scent profiles with interaction term.",
},
"olfactory_adaptation": {
"expr": xi**2/2 - alpha*sp.log(1 + x) + beta*x**2,
"dim": 1,
"category": "perfumery",
"description": "Nose fatigue — logarithmic desensitization to persistent odorants.",
},
}
# =====================================================================
# 92. DREAM DYNAMICS
# =====================================================================
H_DREAM = {
"rem_cycle": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/90)),
"dim": 1,
"category": "dream",
"description": "REM sleep cycle — 90-minute ultradian rhythm as periodic potential.",
},
"latent_narrative": {
"expr": (xi**2 + eta**2)/2 + alpha*sp.exp(-beta*(x**2 + y**2)) * sp.cos(gamma*x),
"dim": 2,
"category": "dream",
"description": "Latent narrative flow — associative memory landscape with modulation.",
},
"dream_instability": {
"expr": xi**2/2 + alpha*x**2 - beta*x**4,
"dim": 1,
"category": "dream",
"description": "Bistable dream state — abrupt transitions between narrative modes.",
},
"hypnagogic_potential": {
"expr": xi**2/2 - alpha*sp.log(sp.cosh(x)) + beta*sp.sin(gamma*x),
"dim": 1,
"category": "dream",
"description": "Hypnagogic state — noise-driven symbolic emergence at sleep onset.",
},
}
# =====================================================================
# 93. GARDENING
# =====================================================================
H_GARDENING = {
"growth_rhythm": {
"expr": xi**2/2 + alpha*x**2*sp.exp(-beta*x),
"dim": 1,
"category": "gardening",
"description": "Plant growth rhythm — sigmoidal biomass accumulation with senescence.",
},
"companion_planting": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.exp(-beta*(x - y)**2) + gamma*(x**2 + y**2),
"dim": 2,
"category": "gardening",
"description": "Companion planting — mutualistic attraction between crop species.",
},
"phototropism_potential": {
"expr": (xi**2 + eta**2)/2 - V0*sp.exp(-alpha*(x - beta)**2),
"dim": 2,
"category": "gardening",
"description": "Phototropism — directional growth toward light source at x=β.",
},
"seasonal_cycle": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/365)),
"dim": 1,
"category": "gardening",
"description": "Annual seasonal cycle — planting/harvesting rhythm over 365 days.",
},
}
# =====================================================================
# 94. TYPOGRAPHY
# =====================================================================
H_TYPOGRAPHY = {
"visual_tension": {
"expr": (xi**2 + eta**2)/2 + alpha*(1 - sp.cos(x - y)),
"dim": 2,
"category": "typography",
"description": "Visual tension between glyph positions — harmonic alignment principle.",
},
"glyph_rhythm": {
"expr": xi**2/2 + V0*sp.sin(alpha*x)*sp.sin(beta*x),
"dim": 1,
"category": "typography",
"description": "Rhythm of glyph spacing — beat frequency in text layout.",
},
"kerning_potential": {
"expr": xi**2/2 + alpha/(sp.Abs(x) + eps) - beta*sp.exp(-gamma*sp.Abs(x)),
"dim": 1,
"category": "typography",
"description": "Kerning dynamics — repulsion at close spacing, attraction at medium range.",
},
"typographic_balance": {
"expr": (xi**2 + eta**2)/2 + alpha*x**2 + beta*y**2 - gamma*x*y,
"dim": 2,
"category": "typography",
"description": "Page layout balance — weighted composition of type elements.",
},
}
# =====================================================================
# 95. CEREMONY
# =====================================================================
H_CEREMONY = {
"ritual_timing": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/T)),
"dim": 1,
"category": "ceremony",
"description": "Ritual timing — periodic structure of ceremonial acts (T = cycle length).",
},
"symbolic_energy": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.log(sp.sqrt(x**2 + y**2) + eps),
"dim": 2,
"category": "ceremony",
"description": "Symbolic energy — focus toward ritual center (e.g., altar, fire).",
},
"liminal_transition": {
"expr": xi**2/2 + alpha*sp.tanh(beta*x),
"dim": 1,
"category": "ceremony",
"description": "Liminal phase — smooth transition between social states (van Gennep).",
},
"communal_synchrony": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.cos(x - y),
"dim": 2,
"category": "ceremony",
"description": "Communal synchrony — phase alignment in group ritual (e.g., chanting, dance).",
},
}
# =====================================================================
# 59. MYTHOPOETICS
# =====================================================================
H_MYTHOPOETICS = {
"hero_journey": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/3)),
"dim": 1,
"category": "mythopoetics",
"description": "Hero's journey — three-act structure as periodic potential.",
},
"threshold_crossing": {
"expr": xi**2/2 - alpha*sp.log(sp.Abs(x) + eps) + beta*x**2,
"dim": 1,
"category": "mythopoetics",
"description": "Liminal threshold — logarithmic barrier between worlds.",
},
"archetypal_duality": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.cos(x - y),
"dim": 2,
"category": "mythopoetics",
"description": "Shadow–self duality — phase alignment of opposing archetypes.",
},
"mythic_recurrence": {
"expr": xi**2/2 + alpha*sp.sin(beta*x)*sp.sin(gamma*x),
"dim": 1,
"category": "mythopoetics",
"description": "Eternal return — interference of nested mythic cycles.",
},
}
# =====================================================================
# 60. CULINARY ARTS
# =====================================================================
H_CULINARY = {
"plating_geometry": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2) + beta*(x*y)**2,
"dim": 2,
"category": "culinary",
"description": "Plating symmetry — visual balance on the plate.",
},
"taste_sequencing": {
"expr": xi**2/2 + V0*sp.exp(-alpha*x)*sp.sin(beta*x),
"dim": 1,
"category": "culinary",
"description": "Taste sequence — transient flavor arc over time.",
},
"umami_resonance": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.exp(-beta*(x - y)**2),
"dim": 2,
"category": "culinary",
"description": "Umami pairing — attraction between complementary tastes.",
},
"bitter_sweet_tension": {
"expr": xi**2/2 + alpha*x**2 - beta*x**3,
"dim": 1,
"category": "culinary",
"description": "Bitter-sweet contrast — cubic instability in flavor profile.",
},
}
# =====================================================================
# 61. DANCE
# =====================================================================
H_DANCE = {
"kinetic_flow": {
"expr": (xi**2 + eta**2)/2 + alpha*(x**2 + y**2),
"dim": 2,
"category": "dance",
"description": "Kinetic energy envelope — bounding ellipse of movement.",
},
"choreographic_potential": {
"expr": (xi**2 + eta**2)/2 + V0*(sp.cos(x) + sp.cos(y) + sp.cos(x + y)),
"dim": 2,
"category": "dance",
"description": "Choreographic lattice — spatial motifs on triangular grid.",
},
"rhythmic_tension": {
"expr": xi**2/2 + alpha*(1 - sp.cos(2*sp.pi*x/T)),
"dim": 1,
"category": "dance",
"description": "Metric pulse — beat-driven potential (T = bar length).",
},
"partner_synchrony": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.cos(x - y),
"dim": 2,
"category": "dance",
"description": "Duet synchrony — phase locking in partner dance.",
},
}
# =====================================================================
# 62. POETICS
# =====================================================================
H_POETICS = {
"metrical_potential": {
"expr": xi**2/2 + V0*(1 - sp.cos(2*sp.pi*x/5)),
"dim": 1,
"category": "poetics",
"description": "Iambic pentameter — 5-beat periodic structure.",
},
"rhyme_attraction": {
"expr": (xi**2 + eta**2)/2 - alpha*sp.exp(-beta*(x - y)**2),
"dim": 2,
"category": "poetics",
"description": "Rhyme coupling — Gaussian attraction between phonetic endpoints.",
},
"caesura_tension": {
"expr": xi**2/2 + alpha*sp.Abs(x - 0.5) - beta*sp.exp(-gamma*(x - 0.5)**2),
"dim": 1,
"category": "poetics",
"description": "Caesura break — linear tension with localized relaxation.",
},
"enjambment_flow": {
"expr": xi**2/2 - alpha*sp.log(sp.Abs(x - 1) + eps),
"dim": 1,
"category": "poetics",
"description": "Enjambment — logarithmic pull across line boundary at x=1.",
},
}
# =====================================================================
# 100. METAPHYSICAL & SPECULATIVE DYNAMICS
# =====================================================================
H_METAPHYSICAL = {
"observer_effect": {
"expr": xi**2/(2*m) + V0*x**2/2 + alpha*sp.Abs(xi)*sp.Abs(x),
"dim": 1,
"category": "metaphysical",
"description": "Observer effect — measurement coupling between position and momentum.",
},
"arrow_of_time": {
"expr": xi**2/(2*m) + k*x**2/2 + beta*sp.exp(-gamma*sp.Abs(xi)),
"dim": 1,
"category": "metaphysical",
"description": "Thermodynamic arrow — irreversible friction in phase space.",
},
"platonian_form": {
"expr": (xi**2 + eta**2)/2 + V0*(x**2 + y**2 - 1)**2 + alpha*(x**2 - y**2)**2,
"dim": 2,
"category": "metaphysical",
"description": "Platonic ideal — perfect symmetry (square + circle) as attractor.",
},
"consciousness_potential": {
"expr": xi**2/2 - alpha*sp.log(sp.cosh(x)) + beta*sp.sin(gamma*x),
"dim": 1,
"category": "metaphysical",
"description": "Integrated information analog — bistable awareness landscape.",
},
"void_dynamics": {
"expr": sp.Abs(xi)**alpha + eps*sp.log(sp.Abs(x) + eps),
"dim": 1,
"category": "metaphysical",
"description": "Dynamics of nothingness — minimal structure emerging from noise.",
},
}
# =====================================================================
# Merge all families
# =====================================================================
CATALOG = {}
for d in [
H_INTEGRABLE, H_CHAOTIC, H_MAGNETIC, H_OPTICAL, H_RELATIVISTIC,
H_POTENTIALS, H_GEOMETRIC, H_QUANTUM, H_ASTROPHYSICS, H_LATTICE,
H_DISSIPATIVE, H_BIOPHYSICS, H_PLASMA, H_ACCELERATOR, H_EXOTIC,
H_CLASSICAL_EXTENDED, H_TOPOLOGICAL, H_NONLINEAR_OPTICS, H_SPIN_SYSTEMS,
H_REACTION_DIFFUSION, H_ELASTICITY, H_STATISTICAL, H_NEUROSCIENCE,
H_ECONOPHYSICS, H_QFT, H_MATHEMATICAL, H_COSMOLOGY, H_TURBULENCE,
H_GRANULAR, H_ACTIVE_MATTER, H_METAMATERIALS, H_QUANTUM_INFO, H_GEOPHYSICS,
H_CLIMATE, H_CAVITY_QED, H_DEFECTS, H_ULTRACOLD, H_STOCHASTIC,
H_STRING_THEORY, H_PARTICLE_PHYSICS, H_QUANTUM_GRAVITY,
H_INTEGRABLE_ADVANCED, H_NON_EQUILIBRIUM, H_TWISTOR, H_SUPERSYMMETRY,
H_DARK_SECTOR, H_NEUTRINO, H_EXOTIC_MATTER, H_QUANTUM_INFO_ADVANCED,
H_PURE_MATH, H_BSM, H_BLACK_HOLES, H_FIELD_THEORY_PROPER,
H_CONTROL_THEORY, H_ACOUSTICS, H_NETWORK_DYNAMICS,
H_SPIN_GLASS, H_MESOSCOPIC, H_POLYMERS,
H_TFT, H_GEOMETRIC_ADVANCED, H_SYMMETRY_REDUCED,
H_QUANTUM_TOPOLOGICAL_EXTENDED, H_CONTINUUM_SOLITONS,
H_STOCHASTIC_ADVANCED, H_MULTI_SCALE_CHAOS, H_MODERN_EXTENSIONS,
H_GAME_DYNAMICS, H_OPTIMIZATION, H_QUANT_FINANCE,
H_SYMBOLIC_COMPUTATION, H_GENERATIVE_DESIGN,
H_EPIDEMIOLOGY, H_LINGUISTICS, H_ECOLOGY, H_INFERENCE,
H_URBAN, H_COGNITIVE, H_LEGAL, H_ART_MUSIC,
H_EDUCATION, H_RELIGION, H_SPORTS, H_AGRICULTURE,
H_PUBLIC_HEALTH, H_ARCHITECTURE, H_CUISINE, H_FASHION,
H_WELLNESS, H_DIGITAL_CULTURE, H_FOLKLORE, H_PERFUMERY,
H_DREAM, H_GARDENING, H_TYPOGRAPHY, H_CEREMONY,
H_MYTHOPOETICS, H_CULINARY, H_DANCE, H_POETICS,
H_METAPHYSICAL
]:
CATALOG.update(d)