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: M → R 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.
Read more about this topic: Hamiltonian Vector Field
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