Numerical Integration

Bodies have:

• Position p (Vec2).
• Velocity v (Vec2).
• Acceleration a (Vec2).
• Mass m (scalar).

Newton's second law: F = m * a → a = F / m.

To advance the simulation by time dt:

Euler integration (simplest, inaccurate):

Problems: gains energy over time. Orbits spiral outward. Used in cheap games.

Symplectic Euler (semi-implicit, simple, stable):

Better than basic Euler. Used in many physics engines.

Verlet integration:

• Stores previous position instead of velocity.
• Velocity is implicit, recovered as (x_next - x_prev) / (2*dt).
• More numerically stable for constraints, because correcting a position implicitly corrects the velocity.
• The basis of position-based cloth and rope solvers. (Box2D itself integrates with symplectic Euler and sequential impulses, not Verlet.)

Runge-Kutta 4 (RK4):

• Higher order; more accurate.
• Used in scientific simulations.
• 4x as expensive per step.
• Probably overkill for games.

For games: symplectic Euler at fixed timestep is the practical default.

Fixed timestep:

Why fixed?

• Deterministic: same input → same output.
• Stable: timestep can't blow up at variable rates.
• Networking: simulations stay in sync.

Variable dt = simulation drift over time.

Forces:

• Gravity: F = m * g (constant 9.8 m/s² on Earth).
• Spring: F = -k * x (Hooke's law).
• Drag: F = -c * v (proportional to velocity).
• Player input: arbitrary.

Sum forces, then integrate. Reset force to zero after each step.