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 physist and mathematician 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.

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 linear isomorphism

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

between the tangent bundle ${\displaystyle TM}$  and the cotangent bundle ${\displaystyle T^{*}M}$ , 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 ${\displaystyle H:M\to \mathbb {R} }$  determines a unique vector field X_H, called the Hamiltonian vector field with the Hamiltonian H, by requiring that for every vector field Y on M, the identity

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

must hold.

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 ${\displaystyle (q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})}$  on M, in which the symplectic form is expressed as

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

Then the Hamiltonian vector field with Hamiltonian H takes the form

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

where Ω is a 2n by 2n square matrix

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

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 ${\displaystyle X_{H}=-\sum a_{ij}p_{i}\partial /\partial q^{j}.}$ 

Properties

• The assignment ${\displaystyle f\mapsto X_{f}}$  is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.

• Suppose that ${\displaystyle (q^{1},\ldots ,q^{n},p_{1},\ldots ,p_{n})}$  are canonical coordinats on M (see above). Then a curve ${\displaystyle \gamma (t)=(q(t),p(t))}$  is an integral curve of the Hamiltonian vector field X_H if and only if it is a solution of the Hamilton's equations:

${\displaystyle {\dot {q}}^{i}={\frac {\partial H}{\partial p_{i}}}}$ 

${\displaystyle {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}.}$ 

• The Hamiltonian H is constant along the integral curves, that is, ${\displaystyle H(\gamma (t))}$  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.

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

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

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

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

where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g. As a consequence, the Poisson bracket satisfies the Jacobi identity

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

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

References

• Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.