# Hamiltonian vector field

(Redirected from Hamiltonian flow)

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

${\displaystyle \omega :TM\to T^{*}M,}$

between the tangent bundle TM and the cotangent bundle T*M, with the inverse

${\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}$

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 defining for every vector field Y on M,

${\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).}$

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 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]

${\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}$

where d denotes the exterior derivative and denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian H takes the form[3]

${\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,}$

where Ω is a 2n × 2n square matrix

${\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}$

and

${\displaystyle \mathrm {d} H={\begin{bmatrix}{\frac {\partial H}{\partial q^{i}}}\\{\frac {\partial H}{\partial p_{i}}}\end{bmatrix}}.}$

The matrix Ω is frequently denoted with J.

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

• If ${\displaystyle H=p_{i}}$ then ${\displaystyle X_{H}=\partial /\partial q^{i};}$
• if ${\displaystyle H=q_{i}}$ then ${\displaystyle X_{H}=-\partial /\partial p^{i};}$
• if ${\displaystyle H=1/2\sum (p_{i})^{2}}$ then ${\displaystyle X_{H}=\sum p_{i}\partial /\partial q^{i};}$
• if ${\displaystyle H=1/2\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}}$ then ${\displaystyle X_{H}=-\sum a_{ij}q_{i}\partial /\partial p^{j}.}$

## Properties

• The assignment fXf is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
• Suppose that (q1, ..., qn, p1, ..., pn) are canonical coordinates on M (see above). Then a curve γ(t) = (q(t),p(t)) is an integral curve of the Hamiltonian vector field XH if and only if it is a solution of the Hamilton's equations:[4]
${\displaystyle {\dot {q}}^{i}={\frac {\partial H}{\partial p_{i}}}}$
${\displaystyle {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}.}$
• The Hamiltonian H is constant along the integral curves, because ${\displaystyle \langle dH,{\dot {\gamma }}\rangle =\omega (X_{H}(\gamma ),X_{H}(\gamma ))=0}$. 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 ${\displaystyle {\mathcal {L}}_{X_{H}}\omega =0.}$

## 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

${\displaystyle \{f,g\}=\omega (X_{g},X_{f})=dg(X_{f})={\mathcal {L}}_{X_{f}}g}$

where ${\displaystyle {\mathcal {L}}_{X}}$ denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds:[5]

${\displaystyle X_{\{f,g\}}=[X_{f},X_{g}],}$

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]

${\displaystyle \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0,}$

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).

## Remarks

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

## Notes

1. ^ Lee 2003, Chapter 18.
2. ^ Lee 2003, Chapter 12.
3. ^ Lee 2003, Chapter 18.
4. ^ Lee 2003, Chapter 18.
5. ^ Lee, Chaptter 18.
6. ^ Lee, Chaptter 18.

## 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.