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Hamiltonian Mechanics

While Lagrangian mechanics formulates dynamics in terms of generalized coordinates and their velocities, Hamiltonian mechanics reformulates the same dynamics in terms of coordinates and their conjugate momenta. This leads to a first-order system of equations that naturally emphasizes phase space structure, conservation laws, and time evolution.

The central object in this formulation is the Hamiltonian, defined as the Legendre transform of the Lagrangian with respect to the generalized velocities.

\[ H(q,p,t) = \sum_i p_i \dot{q}_i - L(q,\dot{q},t), \]

where the generalized momenta are given by

\[ p_i = \frac{\partial L}{\partial \dot{q}_i}. \]

For many mechanical systems, the Hamiltonian coincides with the total energy of the system, though this identification is not fundamental to the formalism.

Hamilton’s Equations

The equations of motion in Hamiltonian mechanics are given by Hamilton’s equations,

\[ \dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}. \]

These equations define a first-order dynamical system on phase space. In contrast to Newtonian or Lagrangian formulations, position and momentum are treated as independent variables whose coupled evolution fully determines the motion.

Phase Space and Structure

Hamiltonian dynamics naturally lives on phase space, where trajectories represent the evolution of the full state of the system. The flow generated by Hamilton’s equations preserves phase space volume and exhibits a rich geometric structure that underlies stability, reversibility, and long-term behavior.

This structure makes Hamiltonian mechanics particularly well suited for studying conservative systems, integrability, perturbations, and long-time numerical behavior. It also provides the natural foundation for symplectic integration methods and modern formulations of control and optimization.

Beyond classical mechanics, the Hamiltonian framework extends directly to statistical mechanics, quantum mechanics, and field theory, and plays a central role in modern dynamical systems theory.