Phase Space Methods and Hamiltonian Systems

The Hamiltonian formulation of classical mechanics provides the natural bridge between ODE theory and physics. Phase space methods reveal conserved quantities, symmetries, and the geometric structure underlying dynamical systems.

Hamiltonian Formulation

Definition5.6Hamilton's Equations

Given a Hamiltonian $H(q, p, t)$ on phase space $\mathbb{R}^{2n}$ with coordinates $(q_1,\ldots,q_n,p_1,\ldots,p_n)$ , Hamilton's equations are $\dot{q}_i = \frac{\partial H}{\partial p_i}$ , $\dot{p}_i = -\frac{\partial H}{\partial q_i}$ . The flow preserves the symplectic form $\omega = \sum dp_i \wedge dq_i$ (Liouville's theorem: phase space volume is conserved). For $H = T + V = \frac{p^2}{2m} + V(q)$ : Hamilton's equations reduce to Newton's second law.

ExampleHarmonic Oscillator in Phase Space

For $H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2$ : Hamilton's equations give $\dot{q} = p/m$ , $\dot{p} = -m\omega^2 q$ . Trajectories are ellipses $\frac{p^2}{2mE} + \frac{m\omega^2 q^2}{2E} = 1$ in phase space. The area enclosed is $A = 2\pi E/\omega$ , which is an adiabatic invariant: under slow variation of $\omega$ , $A$ remains approximately constant while $E$ adjusts.

Poisson Brackets and Integrability

Definition5.7Poisson Brackets and Conservation Laws

The Poisson bracket of functions $f, g$ on phase space is $\{f, g\} = \sum_i\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$ . Hamilton's equations become $\dot{f} = \{f, H\} + \partial f/\partial t$ . A function $f$ is a constant of motion iff $\{f, H\} = 0$ (for time-independent $f$ ). The Poisson bracket satisfies: antisymmetry, linearity, Leibniz rule, and the Jacobi identity $\{f, \{g, h\}\} + \text{cyclic} = 0$ .

RemarkLiouville Integrability

A Hamiltonian system with $n$ degrees of freedom is Liouville integrable if it has $n$ independent constants of motion $F_1 = H, F_2, \ldots, F_n$ in involution: $\{F_i, F_j\} = 0$ . The Liouville-Arnold theorem: the phase space fibers into invariant tori $\mathbb{T}^n$ , and the motion is quasi-periodic. Action-angle variables $(I_i, \theta_i)$ linearize the flow: $\dot{I}_i = 0$ , $\dot{\theta}_i = \omega_i(I)$ . Non-integrable systems exhibit chaos (KAM theory describes the transition).