Time Integration and Energy Drift

The integrate step looked like the trivial one: advance $x$ by $v\,h$. The catch is which velocity and position you use, and the wrong choice quietly creates or destroys energy until the simulation either explodes or grinds to a stop.

Four integrators, one pendulum

Take an undamped pendulum, which should conserve energy exactly, and integrate it four ways at the same step $h$:

• Explicit (forward) Euler. Use the current acceleration and velocity for both updates. Simplest possible, and for an oscillator it pumps energy in: the pendulum spirals outward.
• Semi-implicit (symplectic) Euler. Update velocity first, then use the new velocity for position. One character of difference in code, but it is symplectic, so the energy error stays bounded forever.¹ This is why it is the default in most game and robotics engines.
• Implicit (backward) Euler. Solve for the end-of-step state. Unconditionally stable but dissipative: energy bleeds away and motion damps. Ideal for stiff systems where you want stability, wrong if you care about liveliness.
• RK4. Four derivative evaluations, fourth-order accurate, energy nearly conserved over these timescales, but it costs four times as much and still is not symplectic.

Watch the total energy. Push $dt$ up to exaggerate the split:

What engines actually pick

Most rigid-body engines use semi-implicit Euler: it is cheap and its energy stays bounded, which is exactly the trade a real-time simulator wants. To cut the error further they shrink the effective step with substepping (PhysX's TGS is literally a substepped solver). Stiff or soft systems lean toward implicit integration for its stability. And when conservation over long horizons truly matters, variational integrators derive the update from a discrete action principle, preserving momentum and symplectic structure by construction.²

Integration assumed we already had the forces and the contacts. The next chapter is where the loop finds contacts, and the three after that are where it computes the forces.