Simulation

Symplectic Integration

Symplectic integration is a class of numerical methods for Hamiltonian systems that exactly preserve the symplectic structure of phase space, yielding bounded long-term energy error instead of the systematic drift produced by non-symplectic schemes like explicit Euler. The semi-implicit (symplectic) Euler method, which updates velocity before position, is the workhorse of physics engines including MuJoCo's default integrator. Higher-order members include the Verlet and leapfrog families.

Why it matters for physical AI

Simulated training runs billions of timesteps, so integrators that stay stable at large steps without energy explosion directly determine simulation throughput and the physical plausibility of learned behaviors.