Nbody

Nbody refers to the N-body problem, the challenge of predicting the motions of a system of N bodies that interact through mutual forces, typically gravity. Each body accelerates according to the combined influence of all others, described by a set of coupled differential equations. A common form for gravitational interactions is d^2 r_i/dt^2 = G ∑_{j≠i} m_j (r_j − r_i)/|r_j − r_i|^3, where r_i is the position of body i, m_i its mass, and G the gravitational constant. Initial conditions of positions and velocities determine the subsequent evolution.

Analytical solutions exist only for very small N; for N = 2 closed-form solutions are available, while for N ≥ 3 the problem is generally non-integrable and can exhibit chaotic behavior, with small changes in initial conditions leading to large differences over time. Consequently, numerical methods are essential for simulating realistic systems. Direct N-body computation scales as O(N^2) per timestep, prompting the development of faster algorithms such as Barnes–Hut tree codes (O(N log N)) and fast multipole methods (near O(N)). Common integrators include leapfrog and higher-order Hermite schemes, often augmented with regularization of close encounters or softening to avoid singularities.

Applications span astrophysics, cosmology, and related fields, including studies of star clusters, galaxies, dark matter halos, and planetary formation. The problem is categorized as collisional or collisionless, reflecting the frequency of close encounters and energy exchange, which informs modeling choices. Prominent software packages and codes for N-body simulations include NBODY series, with NBODY6 and NBODY7 among the well-known tools, along with GPU-accelerated and scalable implementations. Ongoing work emphasizes accuracy, long-term energy conservation, relativistic effects, and efficient parallelization for large N.