Symplectic Geometry — Hamiltonian Flow & Liouville
Phase space volume preserved: dq∧dp = invariant, ω = Σ dqᵢ∧dpᵢ
Hamiltonian
Nonlinear pendulum
Duffing oscillator
Hénon-Heiles
Kepler (radial)
Particles N:
30
Energy E:
1.5
Nonlinearity ε:
0.1
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Hamilton's equations:
dq/dt = ∂H/∂p, dp/dt = −∂H/∂q
Liouville: d(vol)/dt = 0
Symplectic form: ω = dq∧dp
Poincaré invariant: ∮ p·dq = const
Noether: symmetry ↔ conservation law
Blob area:
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H variance:
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Time:
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Phase:
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