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1. Classical Dynamics

   Linked via "Hamilton's canonical equations"

   $$H(\mathbf{q}, \mathbf{p}, t) = \sumi pi \dot{q}_i - L$$
   where $pi$ are the generalized momenta, $pi = \frac{\partial L}{\partial \dot{q}_i}$. The evolution of the system in phase space is governed by Hamilton's canonical equations:
   $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$