Suppose the free particle moving inside 2D box. $$H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+V$$ where the potential is zero inside the box and infinite outside the box.
It's clear that $p_x,p_y$ are not constant of motion but $p_x^2,p_y^2$ are. And these are in involution with each other so the system should be integrable.
How do you go about solving the problem? Can we write the explicit form of the solution? \begin{align*} \dot{x}&= \frac{p_x}{m}\ \ \&\ \ \dot{y}= \frac{p_y}{m}\\ \dot{p_x}&= -\partial_x V(x,y),\ \ \&\ \ \dot{p_y}= -\partial_yV(x,y) \end{align*} I don't How do you go about writing an explicit solution of these?