Part II: Hamiltonian Mechanics
The twin formulation that replaces second-order equations with first-order ones, reveals the symplectic geometry of phase space, and provides the bridge to quantum mechanics.
Why Hamiltonian Mechanics?
The Lagrangian formulation is powerful, but the Hamiltonian reformulation goes deeper. By treating position and momentum as independent variables on equal footing, Hamilton's equations reveal that mechanics is fundamentally about flows in phase space. The Poisson bracket structure that emerges is the classical ancestor of the quantum commutator, and canonical transformations give us a systematic way to simplify any mechanical problem.
Liouville's theorem — that phase-space volume is conserved — underpins all of statistical mechanics. The Hamilton-Jacobi equation connects mechanics to optics and wave theory. And the KAM theorem reveals the subtle boundary between order and chaos.
Topics in Part II
The Legendre Transform
The mathematical bridge from the Lagrangian to the Hamiltonian. Understand the geometric meaning of replacing \(\dot{q}\) with \(p\) as the independent variable, the role of convexity, the deep analogy with thermodynamic potentials, and the hybrid Routhian formulation.
Hamilton's Equations
The symmetric first-order equations \(\dot{q} = \partial H/\partial p\) and\(\dot{p} = -\partial H/\partial q\). Learn Poisson brackets as the algebraic structure of mechanics, canonical transformations that preserve this structure, and the Hamilton-Jacobi equation as the ultimate simplification tool.
Phase Space & Liouville's Theorem
Phase space is where classical mechanics truly lives. Learn Liouville's theorem (the incompressibility of Hamiltonian flow), Poincare recurrence, canonical invariants, action-angle variables for integrable systems, and the KAM theorem that governs the boundary between integrability and chaos.
Key Equations of Hamiltonian Mechanics
The Hamiltonian
\[H(q, p, t) = \sum_i p_i\dot{q}_i - L\]
Hamilton's Equations
\[\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}\]
Poisson Bracket
\[\{f, g\} = \sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)\]
Liouville's Theorem
\[\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \{\rho, H\} = 0\]