N-body gravitational simulation

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Integrate the mutual gravity of $N$ bodies — planets, stars, dark-matter particles. Symplectic integrators (leapfrog) keep orbits stable for billions of steps; tree codes (Barnes–Hut) and FMM cut the cost from $O(N^2)$ to $O(N\log N)$ or $O(N)$ .

Formulas

\ddot x_i = \sum_{j\ne i} G m_j\,\frac{x_j - x_i}{\norm{x_j - x_i}^3}

The mathematics behind it

Systems of ODEs★★★★★fundamental
$N$ gravitating bodies give $6N$ coupled first-order equations.
Runge–Kutta methods★★★★★frequent
High-order RK is used for accurate short-term orbits; symplectic methods for long-term stability.
Phase space★★★★★advanced
Symplectic integrators preserve phase-space volume (Liouville), which keeps long simulations physically plausible.
Chaos and sensitivity to initial conditions★★★★★frequent
Three or more gravitating bodies are generically chaotic; long-term planetary predictions have finite horizons.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.