Hamiltonian vector field

{{Short description|Vector field defined for any energy function}}

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.{{sfn|Lee|2003|loc=Chapter 18}}

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 $(M,\omega)$ is a symplectic manifold. Since the symplectic form $\omega$ is nondegenerate, it sets up a fiberwise-linear isomorphism

$\omega: TM \to T^*M,$

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

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

Therefore, one-forms on a symplectic manifold $M$ may be identified with vector fields and every differentiable function $H:M\rightarrow\mathbb{R}$ determines a unique vector field $X_H$ , called the Hamiltonian vector field with the Hamiltonian $H$ , by defining for every vector field $Y$ on $M$ ,

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

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 $(q^1,\cdots,q^n,p_1,\cdots,p_n)$ on $M$ , in which the symplectic form is expressed as:{{sfn|Lee|2003|loc=Chapter 12}} $\omega = \sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i,$

where $\operatorname{d}$ denotes the exterior derivative and $\wedge$ denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian $H$ takes the form:{{sfn|Lee|2003|loc=Chapter 18}} $\Chi_H = \left( \frac{\partial H}{\partial p_i},
- \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,$

where $\Omega$ is a $2n\times 2n$ square matrix

$\Omega =
\begin{bmatrix}
0 & I_n \\
-I_n & 0 \\
\end{bmatrix},$

and

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

The matrix $\Omega$ is frequently denoted with $\mathbf{J}$ .

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

If $H = p_i$ then $X_H = \partial/\partial q^i;$
if $H = q_i$ then $X_H = -\partial/\partial p^i;$
if $H = \frac{1}{2} \sum (p_i)^2$ then $X_H = \sum p_i\partial/\partial q^i;$
if $H = \frac{1}{2} \sum a_{ij} q^i q^j, a_{ij} = a_{ji}$ then $X_H = -\sum a_{ij} q_i\partial/\partial p^j.$

Properties

The assignment $f\mapsto X_f$ is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
Suppose that $(q^1,\cdots,q^n,p_1,\cdots,p_n)$ are canonical coordinates on $M$ (see above). Then a curve $\gamma(t)=(q(t),p(t))$ is an integral curve of the Hamiltonian vector field $X_H$ if and only if it is a solution of Hamilton's equations:{{sfn|Lee|2003|loc=Chapter 18}} $\begin{align}$

\dot{q}^i & = \frac{\partial H}{\partial p_i} \\ \dot{p}_i & =-\frac{\partial H}{\partial q^i}. \end{align}

The Hamiltonian $H$ is constant along the integral curves, because $\langle dH, \dot{\gamma}\rangle = \omega(X_H(\gamma),X_H(\gamma)) = 0$ . That is, $H(\gamma(t))$ 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.See {{harvtxt|Lee|2003|loc=Chapter 18}} for a very concise statement and proof of Noether's theorem.
The symplectic form $\omega$ is preserved by the Hamiltonian flow. Equivalently, the Lie derivative $\mathcal{L}_{X_H} \omega=0$ .

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

$\{f,g\} = \omega(X_g, X_f)= dg(X_f) = \mathcal{L}_{X_f} g$

where $\mathcal{L}_X$ denotes the Lie derivative along a vector field $X$ . Moreover, one can check that the following identity holds:{{sfn|Lee|2003|loc=Chapter 18}} $X_{\{f,g\}}= -[X_f,X_g]$ ,

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:{{sfn|Lee|2003|loc=Chapter 18}} $\{\{f,g\},h\} + \{\{g,h\},f\} + \{\{h,f\},g\}=0$ ,

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

Remarks

Notes

Works cited

{{cite book| title = Foundations of Mechanics

| last1 = Abraham | first1 = Ralph

| last2 = Marsden | first2 = Jerrold E.

| author1-link = Ralph Abraham (mathematician)

| author2-link = Jerrold E. Marsden

| year = 1978

| publisher = Benjamin-Cummings | location = London

| isbn = 978-080530102-1

}}See section 3.2.

{{cite book| title = Mathematical Methods of Classical Mechanics

| last = Arnol'd | first = V.I. | year = 1997

| author-link = Vladimir Arnold

| publisher = Springer | location = Berlin etc

| url = https://archive.org/details/mathematicalmeth0000arno | url-access = registration

| isbn = 0-387-96890-3

}}

{{cite book| title = The Geometry of Physics

| last = Frankel | first = Theodore | year = 1997

| publisher = Cambridge University Press

| url = https://archive.org/details/geometryofphysic0000fran | url-access = registration

| isbn = 0-521-38753-1

}}

{{citation| title = Introduction to Smooth manifolds

| last = Lee | first = J. M. | year = 2003

| volume = 218 | series = Springer Graduate Texts in Mathematics

| isbn = 0-387-95448-1

}}

{{cite book| title = Introduction to Symplectic Topology

| last1 = McDuff | first1 = Dusa

| last2 = Salamon | first2 = D.

| author1-link = Dusa McDuff

| year = 1998

| series = Oxford Mathematical Monographs

| isbn = 0-19-850451-9

}}

External links

Hamiltonian vector field on nLab

Category:Hamiltonian mechanics

Category:Symplectic geometry

Category:William Rowan Hamilton