Symbolic Mechanics / Computational Geometry

Computer Algebra Meets Hamiltonian Geometry

Symbolic computation for deriving, simplifying, and visualizing Hamiltonian, metriplectic, and Lie-Poisson dynamical systems.

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Computer Algebra Meets Hamiltonian Geometry visual summary with symbolic equations, phase portraits, and Casimir leaves

Overview

From symbolic derivation to geometric understanding

This project connects computer algebra with Hamiltonian mechanics to make complex dynamical systems easier to derive, inspect, and visualize. The workflow uses symbolic manipulation to compute Poisson brackets, detect Casimirs, simplify vector fields, characterize equilibria, and prepare structured visualizations of the resulting phase geometry.

ẋ = {x, H}

Symbolic Dynamics

Derive equations of motion directly from assumptions, Hamiltonians, brackets, and coordinate choices.

{C, f} = 0

Geometric Structure

Expose conserved quantities, degeneracies, symmetries, and dissipative extensions before plotting.

∇H · XH = 0

Automated Visualization

Turn symbolic reductions into phase portraits, equilibrium diagrams, and geometric level sets.

Systems Studied

Canonical examples across mechanics and geometry

Spring

Lie group / phase space: T*ℝ
Hamiltonian preview: H = ½(p² + ω²q²)
Dynamics preview: q̇ = p, ṗ = −ω²q
Casimir behavior: Conservative energy contours

Damped Spring

Lie group / phase space: ℝ² with metric term
Hamiltonian preview: H = ½(p² + ω²q²)
Dynamics preview: ṗ = −kq − γp
Casimir behavior: Energy decay toward sink

Pendulum

Lie group / phase space: T*S¹
Hamiltonian preview: H = p²/(2ml²) + mgl(1 − cos θ)
Dynamics preview: θ̇ = ∂H/∂p
Casimir behavior: Separatrix structure

Windy Pendulum

Lie group / phase space: Forced T*S¹
Hamiltonian preview: H(θ, p, t)
Dynamics preview: ṗ = −∂V/∂θ + Fwind
Casimir behavior: Broken conservative foliation

3D Rotating Body

Lie group / phase space: so(3)∗
Hamiltonian preview: H = ½M · I⁻¹M
Dynamics preview: Ṁ = M × ω
Casimir behavior: ||M||² sphere

3D Rigid Body Motion

Lie group / phase space: SE(3)∗
Hamiltonian preview: H = Tᵣₒₜ + Tₜᵣₐₙₛ
Dynamics preview: Lie-Poisson evolution
Casimir behavior: Momentum invariants

Method Pipeline

A symbolic-to-geometric computation path

System Assumptions

coordinates, parameters

Hamiltonian

H(q, p)

Poisson Bracket

{f, g}

Dynamics

X_H

Casimirs

C_i

Equilibria

∇H = 0

Phase Portraits

flow + leaves

Visual Results

Derived portraits and invariant geometry from the paper

**Phase portrait comparisons**Conservative, damped, pendulum, and forced pendulum portraits derived from symbolic dynamics.

**Symbolic-to-geometric workflow**Computer algebra carries assumptions through Hamiltonians, brackets, Casimirs, topology, and visualization.

**Rotating body Casimir sphere**Rigid-body flow constrained to invariant angular momentum leaves.

**Hamiltonian vs. metriplectic behavior**Side-by-side comparison of conservative flow and dissipative bracket structure.

**Hamiltonian geometry workspace**Background mathematical structures used across derivations and visualizations.

Key Mathematical Ideas

Compact reference for the structures used in the paper

XH = J∇H

Hamiltonian Systems

A Hamiltonian packages energy and constraints into a scalar generator whose derivatives define motion.

Code / Reproducibility

Scripts for symbolic derivation and visualization

The computation scripts derive dynamics, characterize equilibria, identify structure-preserving quantities, and generate visualization-ready phase data.

spring_dynamics.mw
pendulum_dynamics.mw
rotating_body_dynamics.mw

derive_hamiltonian()
compute_poisson_bracket()
solve_equilibria()
render_phase_portrait()

Citation

Computer Algebra Meets Hamiltonian Geometry

@article{bajaj2026computer_algebra_hamiltonian,
  title   = {Computer Algebra Meets Hamiltonian Geometry},
  author  = {Bajaj, Chandrajit},
  journal = {Maple Transactions},
  year    = {2026}
}

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