N-Body Gravitational Simulator

N-Body Gravitational Dynamics

This simulation demonstrates the gravitational interactions between multiple bodies in a planar system. Each body attracts every other body according to Newton's law of universal gravitation: F = G m₁ m₂ / r²

Numerical Integration

The simulation uses the Runge-Kutta 4th order (RK4) method to integrate the equations of motion. This provides a good balance between accuracy and computational efficiency for orbital mechanics. For very long-term stability, symplectic integrators like Verlet would be preferred, but RK4 is sufficient for the timescales shown here.

Special Configurations

• Figure-8 Orbit: A remarkable periodic solution discovered by Moore (1993) where three equal masses chase each other in a figure-8 pattern. This demonstrates that chaos is not inevitable in the 3-body problem.
• Binary Stars: Two equal-mass stars orbiting their common center of mass in circular orbits. This is a stable configuration that demonstrates conservation of angular momentum.
• Triple System: A hierarchical triple where a close binary is orbited by a distant third body. This configuration is common in nature and can be stable over long timescales.
• Lagrange Points: A restricted 3-body problem showing the stable L4 and L5 points where a test mass can orbit in synchrony with two larger bodies (like Jupiter's Trojan asteroids).
• Chaotic: A configuration that leads to chaotic motion with sensitive dependence on initial conditions. Small perturbations grow exponentially over time.

Conservation Laws

In an isolated N-body system with only gravitational forces:

• Energy: Total energy (kinetic + potential) is conserved. Deviations indicate numerical errors.
• Momentum: Total linear momentum is conserved, so the center of mass moves at constant velocity.
• Angular Momentum: Total angular momentum is conserved, reflected in the system's overall rotation.

The Three-Body Problem

While the two-body problem has a closed-form solution (Kepler orbits), the three-body problem generally does not. Henri Poincaré proved in the 1890s that the general three-body problem is non-integrable, meaning there's no simple formula for the motion. This was one of the first demonstrations of deterministic chaos in classical mechanics.