cotangent_bundle

cotangent_bundle.py — Symplectic geometry of cotangent bundles: from canonical structure to quantization

Overview

This module provides a symbolic‑numerical laboratory for exploring the symplectic geometry of cotangent bundles before a Hamiltonian is chosen. It demonstrates how the canonical symplectic form ω = dξ ∧ dx emerges directly from the geometry of T*M, and how it alone determines the Poisson bracket, Hamilton’s equations, and ultimately the link to quantum mechanics.

The package implements:

• Construction of the canonical Liouville 1‑form θ and the symplectic form ω = dθ.

• Verification that ω is closed (dω = 0) and non‑degenerate (det ω ≠ 0).

• Derivation of Hamiltonian vector fields X_H from any symbolic Hamiltonian H(x,ξ) via the intrinsic relation ι_{X_H} ω = dH.

• Numerical integration of the Hamiltonian flow with preservation checks (energy, symplectic area, Jacobian determinant).

• Symbolic and numerical study of Poisson brackets, Jacobi identity, and canonical commutation relations.

• Visualisation of phase‑space trajectories, energy surfaces, and the symplectic area preserved by the flow.

• Canonical transformations and their characterisation (preservation of ω).

• A gentle introduction to the Dirac quantisation rule: {f,g} ↦ (1/iℏ)[f̂,ĝ], illustrated with pseudo‑differential operators (through an external psiop module).

All computations are performed symbolically with SymPy and then lambdified for efficient numerical integration and visualisation with NumPy/SciPy and Matplotlib.

Mathematical background

Let M be an n‑dimensional manifold (here taken as ℝⁿ for simplicity). The cotangent bundle T*M carries a canonical 1‑form (Liouville form) defined in local coordinates (x¹,…,xⁿ, ξ₁,…,ξₙ) by

θ = ξ₁ dx¹ + … + ξₙ dxⁿ.

Its exterior derivative

ω = dθ = dξ₁ ∧ dx¹ + … + dξₙ ∧ dxⁿ

is a closed (dω = 0) and non‑degenerate 2‑form – the symplectic form. Crucially, ω is independent of any Hamiltonian; it is a geometric property of the bundle itself.

Given a smooth function H : T*M → ℝ (the Hamiltonian), the Hamiltonian vector field X_H is uniquely defined by

ι_{X_H} ω = dH.

In coordinates this yields the familiar Hamilton equations

˙xⁱ = ∂H/∂ξᵢ ,  ˙ξᵢ = −∂H/∂xⁱ.

The flow Φ_t generated by X_H preserves ω (Liouville’s theorem); in particular the symplectic volume (area in 1D) is conserved. The Poisson bracket of two observables f,g is defined by

{f,g} = ω(X_f, X_g) = Σᵢ (∂f/∂xⁱ ∂g/∂ξᵢ − ∂f/∂ξᵢ ∂g/∂xⁱ),

and satisfies the Jacobi identity. The canonical brackets

{xⁱ, ξⱼ} = δⁱⱼ ,  {xⁱ, xʲ} = {ξᵢ, ξⱼ} = 0

are direct consequences of ω.

Quantisation. The Dirac correspondence replaces Poisson brackets by commutators:

[f̂,ĝ] = iℏ {f,g} .

For the fundamental pair this gives the Heisenberg relation [x̂ⁱ, p̂ⱼ] = iℏ δⁱⱼ. The module illustrates this by quantising a classical Hamiltonian into a pseudo‑differential operator (using a companion module psiop) and verifying the correspondence.

Canonical transformations. A change of coordinates (x,ξ) → (X,Ξ) is canonical iff it preserves ω, i.e. dΞ ∧ dX = dξ ∧ dx. Equivalently, the Jacobian matrix J satisfies Jᵀ ω J = ω. Several examples are provided, together with generating functions.

Higher dimensions. The code handles arbitrary dimension by building the block‑diagonal symplectic matrix

ω = [[0, -I], [I, 0]] .

References

[1]

Abraham, R. & Marsden, J. E. Foundations of Mechanics, Addison‑Wesley, 1978 (2nd ed.). Chapter 3: Hamiltonian Systems.

[2]

Arnold, V. I. Mathematical Methods of Classical Mechanics, Springer‑Verlag, 1989 (2nd ed.). §37: Symplectic Structure.

[3]

Cannas da Silva, A. Lectures on Symplectic Geometry, Springer‑Verlag, 2001. Chapter 1: Symplectic Forms.

[4]

Dirac, P. A. M. The Principles of Quantum Mechanics, Oxford University Press, 1930 (4th ed.). §21: Poisson Brackets.

[5]

McDuff, D. & Salamon, D. Introduction to Symplectic Topology, Oxford University Press, 2017 (3rd ed.). Chapter 1: Symplectic Manifolds.

[6]

Weyl, H. The Theory of Groups and Quantum Mechanics, Dover Publications, 1931 (translated from German). §II.7: Quantisation and the Symplectic Form.

cotangent_bundle.canonical_symplectic_form_construction()

Step-by-step construction of the canonical symplectic form on the cotangent bundle T*M

cotangent_bundle.canonical_transformations_preserve_omega()

Shows that canonical transformations preserve ω

cotangent_bundle.complete_symplectic_example_unified()

Complete example showing the ENTIRE pipeline: H → ω → X_H → Φₜ → Conserved quantities

Uses classes from psiop.py for integration

cotangent_bundle.conceptual_architecture()

High-level overview of the conceptual architecture

cotangent_bundle.explicit_derivation_hamilton_equations()

Explicitly derives Hamilton’s equations from only the data of ω and H

cotangent_bundle.from_omega_to_flow(H_expr, vars_x)

Shows how the symplectic structure ω allows passage from H (function) to X_H (vector field) to Φₜ (flow)

cotangent_bundle.from_poisson_to_commutator()

Shows the fundamental link between Poisson bracket (classical) and commutator (quantum) via correspondence

cotangent_bundle.hamiltonian_vector_field(H_expr, vars_x)

Computes the Hamiltonian vector field X_H from an arbitrary H

cotangent_bundle.interactive_symplectic_dynamics(H_options)

Interactive dashboard to explore different Hamiltonians with the SAME symplectic structure

cotangent_bundle.poisson_bracket(f_expr, g_expr, vars_x)

Poisson bracket {f, g} induced by ω

cotangent_bundle.quantize_hamiltonian(H_classical, vars_x, mode='weyl')

Quantizes a classical Hamiltonian into a pseudo-differential operator

H_classical(x, ξ) → Ĥ operator

mode: ‘weyl’ (symmetric) or ‘kn’ (Kohn-Nirenberg)

cotangent_bundle.symplectic_structure_higher_dimensions(n=2)

Shows how ω generalizes to arbitrary dimension

cotangent_bundle.verify_canonical_relations(vars_x)

Verifies the canonical commutation relations

cotangent_bundle.verify_symplectic_preservation(H_expr, vars_x, t_max=1.0, n_times=10)

Simplified and more robust version

cotangent_bundle.verify_symplectic_structure(dim=1)

Verify that ω is closed and non-degenerate

cotangent_bundle.visualize_symplectic_geometry(H_expr, vars_x, xlim=(-2, 2), klim=(-2, 2))

Visualizes the symplectic geometry for an arbitrary H