Every massive object in the universe pulls on every other via gravity: F = Gm₁m₂/r². An N-body simulation tracks how N objects move under each other's gravity over time. The challenge: we have no closed-form solution for N ≥ 3. We must step forward numerically — integrate the equations of motion.

Real use-cases: predicting planetary orbits, simulating star cluster evolution, modelling galaxy mergers, designing spacecraft trajectories.

Order: 2nd order accuracy. Best for: long orbital simulations — planetary orbits, galaxy evolution.

The leapfrog splits the update: velocities are computed at half-steps, positions at full steps — they "leap" over each other. This makes it symplectic: it conserves a modified Hamiltonian, so total energy doesn't drift over millions of steps.

Used in: GADGET (cosmological N-body code), most planetary mechanics codes.

Order: 4th order accuracy. Best for: short simulations, educational demos, smooth trajectories.

RK4 takes 4 sample "slopes" (k₁–k₄) at different points within each timestep, then combines them in a weighted average. Highly accurate per step but NOT symplectic — energy drifts slowly over very long runs.

Used in: spacecraft trajectory design, short-term orbit prediction, general ODE solving.

Order: 4th order. Best for: dense stellar clusters, close encounters, professional stellar dynamics.

Hermite uses both acceleration and its time derivative (the "jerk") — how quickly force is changing. It predicts a position, evaluates the new force + jerk, then corrects. This makes it excellent near close encounters where forces change rapidly.

Used in: NBODY6, NBODY7 (Aarseth codes) — the gold standard for globular cluster simulations.

Complexity: O(N log N) vs O(N²) brute force. Best for: large N (hundreds to millions of particles), galaxy simulations.

For N bodies, brute force needs N² force calculations per step. Barnes-Hut builds an octree (quadtree in 2D) — distant clusters are approximated as a single body at their centre of mass. If s/d < θ (opening angle ≈ 0.5), treat the whole node as one particle.

Used in: GADGET-2, Millennium Simulation, most large-scale cosmological codes.

☀️ Solar System — Sun + 4 planets (Mercury→Mars) with real orbital speeds. Leapfrog runs stably for thousands of orbits.

⭐ Binary Star — Two equal-mass stars in mutual orbit. Add a third body and watch chaos emerge (restricted 3-body problem).

🌀 Galaxy Collision — Two disc galaxies passing through each other; tidal streams and merger dynamics.

⬛ Black Hole — Massive central body with orbiting stars; watch tidal disruption and slingshot events.

♾️ Figure-8 — Exact periodic solution for 3 equal masses. Beautiful, fragile, mathematically proven to exist in 2000.

🎲 Random N-Body — Set N and watch chaos unfold. Try 50–200 bodies with Barnes-Hut.