Numerical Integration: Verlet vs. Euler Physics Simulation

When building physics engines for video games, graphics simulations, or astronomical modeling, computer hardware cannot continuously integrate physical differential equations. Instead, physics engines approximate motion using numerical integration over discrete time steps $\Delta t$.

1. Summary & Key Takeaways

• Explicit Euler Method: Updates position using current velocity: $x_{t+\Delta t} = x_t + v_t \Delta t, \quad v_{t+\Delta t} = v_t + a_t \Delta t$
  • Flaw: Accumulates truncation error over time, causing physical systems to artificially gain energy and explode!
• Verlet Integration: Evaluates position from current $x_t$ and previous $x_{t-\Delta t}$ frame positions: $x_{t+\Delta t} = 2x_t - x_{t-\Delta t} + a_t \Delta t^2$
  • Advantage: Symplectic & time-reversible, accurately conserving total mechanical energy (Potential + Kinetic).

2. Interactive Physics Simulator

Use the interactive pendulum engine below to toggle between Explicit Euler and Verlet Integration!

// Verlet vs. Explicit Euler Numerical Integration in C++
struct Particle {
    double x, y;
    double oldX, oldY; // For Verlet Integration
    double vx, vy;     // For Euler Integration
    double ax, ay;
};

// 1. Explicit Euler Integration (Unstable for orbits & springs)
void stepEuler(Particle& p, double dt) {
    p.x += p.vx * dt;
    p.y += p.vy * dt;
    p.vx += p.ax * dt;
    p.vy += p.ay * dt;
}

// 2. Verlet Integration (Symplectic & Energy Preserving)
void stepVerlet(Particle& p, double dt) {
    double nextX = 2 * p.x - p.oldX + p.ax * dt * dt;
    double nextY = 2 * p.y - p.oldY + p.ay * dt * dt;
    p.oldX = p.x; p.oldY = p.y;
    p.x = nextX; p.y = nextY;
}

3. Mathematical Foundations

Verlet integration is derived directly by adding the forward and backward Taylor expansions of position $x(t)$:

Forward Taylor Series ($t + \Delta t$)

$x(t + \Delta t) = x(t) + v(t)\Delta t + \frac{1}{2}a(t)\Delta t^2 + \frac{1}{6}b(t)\Delta t^3 + \mathcal{O}(\Delta t^4)$

Backward Taylor Series ($t - \Delta t$)

$x(t - \Delta t) = x(t) - v(t)\Delta t + \frac{1}{2}a(t)\Delta t^2 - \frac{1}{6}b(t)\Delta t^3 + \mathcal{O}(\Delta t^4)$

Adding Both Equations (Eliminates Velocity & Odd Terms!)

$x(t + \Delta t) = 2x(t) - x(t - \Delta t) + a(t)\Delta t^2 + \mathcal{O}(\Delta t^4)$

Because the $\mathcal{O}(\Delta t^3)$ term cancels out, Verlet integration achieves 4th-order local accuracy without calculating velocity vectors directly!

4. Integration Method Comparison Matrix