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

{{Reflist|group=nb}}

Notes

{{Reflist|22em}}

Works cited

{{refbegin}}

  • {{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

}}

{{refend}}