Given H ∈ C^∞(M), the unique vector field X_H determined by ι_{X_H}ω = dH. Flow of X_H preserves both ω (Liouville) and H (energy conservation) — the geometric content of Hamilton’s equations.
Hamiltonian vector field
Related concepts
- Symplectic form
- Poisson bracket
- Arnold conjecture
- Liouville's theorem (Hamiltonian flow)
- Action-angle coordinates
- Arnold-Liouville integrability theorem
- Poincaré–Cartan integral invariant
- Noether theorem (Hamiltonian form)
- Legendre transform (Lagrangian ↔ Hamiltonian)
- Symplectic eigenvalue pairing
- Williamson's normal form
- Hofer metric on Ham(M, ω)
- Hamiltonian mechanics