1. Hamilton's Equations in Real Space

Consider a general Hamiltonian $H(\{x_j\}, \{p_j\})$ for $N$ particles in one dimension. The Hamilton's equations are

$$\dot{x}_j = \frac{\partial H}{\partial p_j}, \quad \dot{p}_j = - \frac{\partial H}{\partial x_j}, \quad j = 0,1,\dots,N-1.$$ (32)