Hamiltonian Systems

Hamiltonian mechanics provides the physical motivation for symplectic geometry. A Hamiltonian system on a symplectic manifold $(M, \omega)$ is determined by a smooth function $H: M \to \mathbb{R}$ , whose flow preserves the symplectic form.

Hamiltonian Vector Fields

Definition4.1Hamiltonian Vector Field

Given a smooth function $H: M \to \mathbb{R}$ on a symplectic manifold $(M, \omega)$ , the Hamiltonian vector field $X_H$ is the unique vector field satisfying $\iota_{X_H} \omega = dH$ , i.e., $\omega(X_H, \cdot) = dH$ . The flow $\phi_t^H$ of $X_H$ is called the Hamiltonian flow of $H$ .

Definition4.2Poisson Bracket

The Poisson bracket of two smooth functions $F, G \in C^\infty(M)$ is $\{F, G\} = \omega(X_F, X_G) = X_G(F) = -X_F(G)$ . In canonical coordinates $(q_i, p_i)$ : $\{F, G\} = \sum_{i=1}^n \left(\frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}\right).$

ExampleHamilton's Equations

In canonical coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ with $\omega = \sum dq_i \wedge dp_i$ , the Hamiltonian vector field gives Hamilton's equations: $\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}.$

Conservation Laws

RemarkEnergy Conservation

Along the Hamiltonian flow, $\frac{d}{dt}H(\phi_t^H(x)) = dH(X_H) = \omega(X_H, X_H) = 0$ . Thus the Hamiltonian function is conserved along its own flow. More generally, $F$ is conserved along $X_H$ if and only if $\{F, H\} = 0$ .

Definition4.3Completely Integrable System

A Hamiltonian system on a $2n$ -dimensional symplectic manifold is completely integrable (in the sense of Liouville-Arnold) if there exist $n$ independent functions $F_1 = H, F_2, \ldots, F_n$ in involution: $\{F_i, F_j\} = 0$ for all $i, j$ . The level sets of $(F_1, \ldots, F_n)$ are then Lagrangian submanifolds.