Lagrangian Sandbox

Analytical mechanics in action. Pick a system and its Euler-Lagrange equations, derived from $L=T-V$, are integrated by RK4 (the verified shared engine). The left panel animates the mechanism; the right is the phase portrait $(q,\dot q)$. The readouts are the conserved quantities Noether's theorem guarantees: the Hamiltonian, because the Lagrangian has no explicit time dependence, and the angular momentum whenever the potential is rotationally symmetric, which the Kepler and gravity-free spring obey but the gravity-loaded pendulum does not. Watch the energy readout sit still while a chaotic double pendulum thrashes.