# 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.[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*.

## 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 *X _{H}*, 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.

## Examples

Suppose that *M* is a 2*n*-dimensional symplectic manifold. Then locally, one may choose canonical coordinates (*q*^{1}, ..., *q ^{n}*,

*p*

_{1}, ...,

*p*) on

_{n}*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[3]

where Ω is a 2*n* × 2*n* square matrix

and

The matrix Ω is frequently denoted with **J**.

Suppose that *M* = **R**^{2n} is the 2*n*-dimensional symplectic vector space with (global) canonical coordinates.

- If then
- if then
- if then
- if then

## Properties

- The assignment
*f*↦*X*is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields._{f} - Suppose that (
*q*^{1}, ...,*q*,^{n}*p*_{1}, ...,*p*) are canonical coordinates on_{n}*M*(see above). Then a curve γ(*t*) =*(q(t),p(t))*is an integral curve of the Hamiltonian vector field*X*if and only if it is a solution of the Hamilton's equations:[4]_{H}

- The Hamiltonian
*H*is constant along the integral curves, because . That is,*H*(γ(*t*)) is actually independent of*t*. This property corresponds to the conservation of energy in Hamiltonian mechanics. - More generally, if two functions
*F*and*H*have a zero Poisson bracket (cf. below), then*F*is constant along the integral curves of*H*, and similarly,*H*is constant along the integral curves of*F*. This fact is the abstract mathematical principle behind Noether's theorem.[nb 1] - The symplectic form ω is preserved by the Hamiltonian flow. Equivalently, the Lie derivative

## 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:[5]

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[6]

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* ↦ *X _{f}* is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if

*M*is connected).

## Remarks

- See Lee (2003, Chapter 18) for a very concise statement and proof of Noether's theorem.

## Notes

## References

- Abraham, Ralph; Marsden, Jerrold E. (1978).
*Foundations of Mechanics*. London: Benjamin-Cummings. ISBN 9780805301021.*See section 3.2*. - Arnol'd, V.I. (1997).
*Mathematical Methods of Classical Mechanics*. Berlin etc: Springer. ISBN 0-387-96890-3. - Frankel, Theodore (1997).
*The Geometry of Physics*. Cambridge: Cambridge University Press. ISBN 0-521-38753-1. - Lee, J. M. (2003),
*Introduction to Smooth manifolds*, Springer Graduate Texts in Mathematics,**218**, ISBN 0-387-95448-1 - McDuff, Dusa; Salamon, D. (1998).
*Introduction to Symplectic Topology*. Oxford Mathematical Monographs. ISBN 0-19-850451-9.