Hamiltonian vector field

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In mathematics and physics, a Hamiltonian vector field is a vector field induced on a Poisson manifold or symplectic manifold by an energy function or Hamiltonian. The integral curves of the symplectic vector field are solutions to the Hamilton–Jacobi equations of motion. The vector field, when taken together with the symplectic manifold and the symplectic form on the manifold, comprise a Hamiltonian system. The symplectomorphisms arising from the flow of a Hamiltonian vector field are known as Hamiltonian symplectomorphisms.

[edit] Definition

Since the symplectic form on a symplectic manifold is nondegenerate, it sets up an isomorphism between the tangent bundle and the cotangent bundle, thus establishing a one-to-one correspondence between tangent vectors and one-forms. As a special case, every differentiable function H:M\to\mathbb{R} on a symplectic manifold M defines a unique vector field, XH, called a Hamiltonian vector field. It is defined such that for every vector field Y on M the identity

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

holds. In canonical coordinates (q^1,\ldots ,q^n,p_1,\ldots,p_n), the symplectic form can be written as

\omega=\sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i

and thus the Hamiltonian vector field takes the form

X_H = \left( \frac{\partial H}{\partial p_i}, - \frac{\partial H}{\partial q^i} \right) = \Omega \cdot \mathrm{d}H

where Ω is the canonical symplectic matrix

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

A curve γ(t) = (q(t),p(t)) is thus an integral curve of the vector field if and only if it is a solution of the Hamilton–Jacobi equations:

\dot{q}^i = \frac {\partial H}{\partial p_i}

and

\dot{p}_i = - \frac {\partial H}{\partial q^i}.

Note that the energy is a constant along the integral curve, that is, H(γ(t)) is a constant independent of t.

[edit] Poisson bracket

The Hamiltonian vector fields give differentiable functions on M the structure of a Lie algebra with bracket the Poisson bracket

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

where \mathcal{L}_X is the Lie derivative along X. Note that some authors use sign conventions that differ from the above.

[edit] 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.
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