Hamiltonian Vector Field - Definition

Definition

Suppose that (M,ω) is a symplectic manifold. Since the symplectic form ω is nondegenerate, it sets up a fiberwise-linear isomorphism

between the tangent bundle TM and the cotangent bundle T*M, with the inverse

Therefore, one-forms on a symplectic manifold M may be identified with vector fields and every differentiable function H: MR determines a unique vector field XH, called the Hamiltonian vector field with the Hamiltonian H, by requiring that for every vector field Y on M, the identity

must hold.

Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.

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