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.
[1] 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: M → R determines a unique vector field XH, called the Hamiltonian vector field with the Hamiltonian H, by defining for every vector field Y on M,
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:[2]
where d denotes the exterior derivative and ∧ denotes the exterior product.
Then the Hamiltonian vector field with Hamiltonian H takes the form:[1]
where Ω is a 2n × 2n square matrix
The matrix Ω is frequently denoted with J.
Suppose that M = R2n is the 2n-dimensional symplectic vector space with (global) canonical coordinates.
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
denotes the Lie derivative along a vector field X.
Moreover, one can check that the following identity holds:[1]
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:[1]
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 f ↦ Xf is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if M is connected).