6.8: Hamilton’s Equations

Having finally established that we can write, for an incremental change along the dynamical path of the system in phase space,

$$d H\left(q_{i}, p_{i}\right)=-\sum_{i} \dot{p}_{i} d q_{i}+\sum_{i} \dot{q}_{i} d p_{i}$$

we have immediately the so-called canonical form of Hamilton’s equations of motion:

$$\begin{align*} \dfrac{\partial H}{\partial p_{i}} &=\dot{q}_{i} \\[4pt] \dfrac{\partial H}{\partial q_{i}} &=-\dot{p}_{i} \end{align*}$$

Evidently going from state space to phase space has replaced the second order Euler-Lagrange equations with this equivalent set of pairs of first order equations.