Physics engines

What is it?

Why does it exist?

Formulas

$$v_{n+1} = v_n + h\,\frac{F_n}{m}, \qquad x_{n+1} = x_n + h\,v_{n+1}$$

semi-implicit (symplectic) Euler

$$x_{n+1} = 2x_n - x_{n-1} + h^2\,\frac{F_n}{m}$$

Verlet integration (second order, symplectic)

How is it computed?

every frame (h = 1/60 s, often split into substeps):
    for each body: F = gravity + springs + user forces
    for each body: v += h · F/m ;  x += h · v        # semi-implicit Euler
    detect collisions;  apply impulses to fix contacts

Why does it matter?

The mathematics behind it

• Vectors★★★★★fundamental

  Positions, velocities and forces are 3D vectors updated every frame.

• Derivative★★★★★fundamental

  Engines store positions and their derivatives (velocities) and advance them frame by frame.

• Numerical stability★★★★★fundamental

  Explicit integrators with too large a time step blow up; engines pick stable schemes and sub-step.

• Ordinary differential equations★★★★★fundamental

  A physics engine integrates $\dot x = v$, $\dot v = F/m$ for every body, every frame.

• Euler's method★★★★★fundamental

  Semi-implicit Euler is the default integrator of Box2D, Bullet and most game engines.

• Definite integral★★★★★frequent

  Positions are integrals of velocities, velocities integrals of accelerations.

• Fundamental theorem of calculus★★★★★fundamental

  Integrating velocity gives position: every simulation step is a tiny application of the FTC.

• Taylor polynomial★★★★★frequent

  Integrators (Euler, Verlet) are truncated Taylor expansions of the motion in the time step.

• Second-order linear equations (oscillations)★★★★★frequent

  Springs and dampers drive cloth, ragdolls, camera smoothing and vehicle suspensions.

• Stiffness and implicit methods★★★★★frequent

  Cloth and soft bodies use implicit integration (Baraff–Witkin) so stiff springs do not explode.

• Continuity★★★★★frequent

  Simulation assumes motion is continuous between frames; collisions break that and are handled separately.

• Newton's method★★★★★advanced

  Implicit integrators for stiff systems (cloth, soft bodies) solve a nonlinear system with Newton each step.

• Vector fields★★★★★frequent

  Wind, gravity wells and particle effects are applied as force fields.

• Initial value problems: existence and uniqueness★★★★★frequent

  Non-Lipschitz forces (contacts, friction switches) break uniqueness, which is why engines treat them specially.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• N-body gravitational simulation
• Fluid dynamics and CFD