Symplectic

A symplectic structure is an antisymmetric non-degenerate 2-form on a manifold. Phase space carries the canonical symplectic form ω = Σ dqᵢ ∧ dpᵢ, and Hamiltonian flow is defined precisely as the flow that preserves this form. Liouville's theorem (conservation of phase-space volume) is an immediate corollary: volume is ω^n / n! for an n-degree-of-freedom system.

Numerical integrators that preserve the symplectic form — leapfrog, velocity-Verlet, Yoshida — are called symplectic integrators. They do not conserve energy exactly, but their energy error is bounded and oscillates rather than drifting. This is why symplectic integrators are standard for long-time orbital simulations: a conventional forward-Euler integrator drifts a planetary orbit outward, but a leapfrog integrator keeps the orbit closed to machine precision for billions of years.