PX285 - F3a - particle in 1D

H = T + V = \frac{1}{2 m} p^{2} + V (x)

\begin{aligned} \dot{x} = \frac{\partial H}{\partial p} & = \frac{p}{m} \\ ⟹ d x & = \frac{p}{m} d t \\ \dot{p} = - \frac{\partial H}{\partial x} & = - \frac{\partial V}{\partial x} = F \\ ⟹ d p & = F d t \end{aligned}

for a constant $V$ , $\frac{d p}{d t} = F = - \frac{\partial V}{\partial x}$ , so $p$ is constant