Hamiltonian Mechanics

Legendre Transform

Define conjugate momenta

$$p_i = \frac{\partial L}{\partial \dot q_i}.$$

The Hamiltonian is

$$H(q,p,t)=\sum_i p_i\dot q_i - L(q,\dot q,t).$$

Hamilton’s Equations

$$\dot q_i = \frac{\partial H}{\partial p_i}, \qquad \dot p_i = -\frac{\partial H}{\partial q_i}.$$

This form makes conservation laws and phase-space geometry visible.