Hamiltonian Vector Field - Properties

Properties

  • The assignment f ↦ Xf is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
  • Suppose that (q1, ..., qn, p1, ..., pn) are canonical coordinates on M (see above). Then a curve γ(t)=(q(t),p(t)) is an integral curve of the Hamiltonian vector field XH if and only if it is a solution of the Hamilton's equations:
  • The Hamiltonian H is constant along the integral curves, that is, H(γ(t)) is actually independent of t. This property corresponds to the conservation of energy in Hamiltonian mechanics.
  • More generally, if two functions F and H have a zero Poisson bracket (cf. below), then F is constant along the integral curves of H, and similarly, H is constant along the integral curves of F. This fact is the abstract mathematical principle behind Noether's theorem.
  • Symplectic form ω is preserved by Hamiltonian flow; or equivalently, Lie derivative

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