In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics. Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of f and g. Definition Suppose that is a symplectic manifold. Since the symplectic form is nondegenerate, it sets up a fiberwise-linear isomorphism between the tangent bundle and the cotangent bundle , with the inverse Therefore, one-forms on a symplectic manifold may be identified with vector fields and every differentiable function determines a unique vector field , called the Hamiltonian vector field with the Hamiltonian , by defining for every vector field on , 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. Examples Suppose that is a -dimensional symplectic manifold. Then locally, one may choose canonical coordinates on , in which the symplectic form is expressed as: where denotes the exterior derivative and denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian takes the form: where is a square matrix and The matrix is frequently denoted with . Suppose that M = R2n is the 2n-dimensional symplectic vector space with (global) canonical coordinates. If then if then if then if then Properties The assignment is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields. Suppose that are canonical coordinates on (see above). Then a curve is an integral curve of the Hamiltonian vector field if and only if it is a solution of Hamilton's equations: The Hamiltonian is constant along the integral curves, because . That is, is actually independent of . This property corresponds to the conservation of energy in Hamiltonian mechanics. More