Hamiltonian vector field

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.


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.


Suppose that M is a 2n-dimensional symplectic manifold. Then locally, one may choose canonical coordinates (q1, ..., qn, p1, ..., pn) on M, in which the symplectic form is expressed as

where d denotes the exterior derivative and ∧ denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian H takes the form

where Ω is a 2n by 2n square matrix


Suppose that M = R2n is the 2n-dimensional symplectic vector space with (global) canonical coordinates.


Poisson bracket

The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold M, the Poisson bracket, defined by the formula

where denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds:

where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity

which means that the vector space of differentiable functions on M, endowed with the Poisson bracket, has the structure of a Lie algebra over R, and the assignment fXf is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if M is connected).


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