Extending Coordinate Objects to High‐Dimensional Phase Spaces and Poisson Brackets

1. Need for High-Dimensional Phase Spaces

In classical mechanics and modern physics, phase spaces are fundamental for describing system states. For an n-degree-of-freedom system: - Phase space is 2n-dimensional (n generalized coordinates qᵢ and n conjugate momenta pᵢ) - Requires extension beyond traditional 3D coordinates to support: - Symplectic geometry (preserving ω = ∑ᵢ dqᵢ ∧ dpᵢ) - Poisson bracket calculations: {F,G} = ∑ᵢ (∂F/∂qᵢ ∂G/∂pᵢ - ∂F/∂pᵢ ∂G/∂qᵢ) - Hamiltonian dynamics representation

2. Phase Space Coordinate Design

Data Structure

struct phase_coord {
    std::vector<vec> q_basis; // Generalized coordinate basis (n-dim)
    std::vector<vec> p_basis;  // Conjugate momentum basis (n-dim)
    vec origin;               // Origin point (2n-dim)
    vec scaling;              // Scaling factors (2n-dim)
};

Key Operations

Multiplication (Coordinate Composition):

phase_coord operator*(const phase_coord& c1, const phase_coord& c2) {
    phase_coord result;
    for (int i = 0; i < n; ++i) {
        result.q_basis[i] = c1.q_basis[i] * c2.q_basis[i] 
                         - c1.p_basis[i] * c2.q_basis[i+n];
        result.p_basis[i] = c1.q_basis[i] * c2.p_basis[i] 
                         + c1.p_basis[i] * c2.p_basis[i+n];
    }
    return result;
}

Division (Coordinate Projection):

phase_coord operator/(const phase_coord& c1, const phase_coord& c2) {
    // Returns c1 * inv(c2) preserving symplectic structure
    ...
}

3. Poisson Bracket Implementation

Differential Projection

vec dF = F / current_coord;  // Gets ∂F/∂qᵢ, ∂F/∂pᵢ components

Symplectic Calculation

double poisson_bracket(const vec& dF, const vec& dG) {
    vec J_dG = symplectic_matrix * dG;  // J = [0 I; -I 0]
    return dot_product(dF, J_dG);
}

Example (Angular Momentum):

phase_coord Lx, Ly, Lz;
double Lz_val = poisson_bracket(Lx / current_coord, Ly / current_coord);
// Satisfies {Lx, Ly} ≈ Lz

4. Connection and Curvature

Connection Calculation

phase_coord Gu = (C1 / C2 - identity_phase_coord()) / epsilon;
// Components correspond to Γᵢⱼᵏ

Curvature Tensor

phase_coord Guv = Gu * Gv - Gv * Gu;  // Riemann curvature R(X,Y)

5. Example: Harmonic Oscillator

Initialization:

phase_coord harm_coord;
harm_coord.q_basis = {vec(1,0), vec(0,1)};  // q1, q2 bases
harm_coord.p_basis = {vec(0,1), vec(-1,0)}; // p1, p2 bases (ω(qi,pj)=δᵢⱼ)

Time Evolution:

double H = 0.5 * (q1*q1 + p1*p1);  // Hamiltonian
vec dH = H / harm_coord;
vec dA = A / harm_coord;
double dA_dt = poisson_bracket(dA, dH);  // Time derivative

6. Summary

Key advantages of this approach:

1. Unifies physics and geometry through coordinate operations

2. Simplifies implementation via operator overloading

3. Broad applicability across classical/quantum systems

The core idea is representing physics through programmable geometric operations, making advanced concepts more accessible for computational implementation.