Pendulum on a moving cart

What you are seeing: a frictionless cart of mass $M$ constrained to move horizontally, with a rigid pendulum of length $L$ and bob mass $m$ hanging from a pivot on top of the cart. Classical two-degree-of-freedom Lagrangian system. Both energy and total horizontal momentum are conserved (no external horizontal forces).

Release the pendulum from angle $\theta_0$ and watch it swing. The cart slides to keep total $p_x = (M+m) \dot{x} + m L \cos\theta \, \dot\theta$ constant. The traces below show energy drift (tiny under RK4) and the $\theta$ vs cart-$x$ phase portrait.