N-body simulation

Builds on n-body problem Newton's law of universal gravitation Ordinary differential equation

Definition

Let $N \ge 2$ and let $G > 0$ be the gravitational constant. Consider point particles with masses $m_1, \dots, m_N > 0$ , positions $x_i(t) \in \R^3$ and a softening length $\varepsilon \ge 0$ . The gravitational N-body equations are the system of ordinary differential equations $\ddot x_i = -G \sum_{j \ne i} m_j \frac{x_i - x_j}{\left(\norm{x_i - x_j}^2 + \varepsilon^2\right)^{3/2}}, \qquad i = 1, \dots, N.$ An N-body simulation is a numerical computation that starts from given positions and velocities at time $t_0$ . By a time-stepping scheme applied to this system, it produces approximations of $x_i(t_k)$ and $\dot x_i(t_k)$ at times $t_0 < t_1 < t_2 < \cdots$ .

Remarks

For $\varepsilon = 0$ the equations are those of the n-body problem with Newton's law of gravitation. A positive $\varepsilon$ removes the divergence of the force at close encounters; it is used when each particle stands for a cloud of matter rather than a star.

Example: the leapfrog (kick-drift-kick) scheme with step $h$ , which updates $v_i \leftarrow v_i + \tfrac{h}{2} a_i$ , then $x_i \leftarrow x_i + h v_i$ , then recomputes $a_i$ and sets $v_i \leftarrow v_i + \tfrac{h}{2} a_i$ , defines an N-body simulation. Each step only uses the right-hand side of the equations, so it is a time-stepping scheme for them.

Non-example: the closed-form Kepler orbit of two bodies is an exact solution, not a step-by-step numerical computation, so it is not an N-body simulation.

Direct summation of all pairs costs of order $N^2$ operations per step. Tree codes and the fast multipole method lower this cost by approximating the forces of distant groups of particles.

Reading path

To understand this page from the ground up, read these 47 pages in order. The foundations come first.

Derivative concept
Differential equation concept
Euclidean space object
Euclidean vector concept
First integral concept
Function concept
Hamiltonian system concept
Interval concept
Ordinary differential equation concept
Velocity concept
Fréchet derivative definition
Cross product definition
Ordinary differential equation definition
Function of several variables concept
Vector-valued function concept
Classical mechanics concept
Lipschitz condition definition
n-body problem problem
Limit of a function of several variables concept
Momentum concept
Partial derivative concept
Solution of an ordinary differential equation definition
Inertial frame of reference concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Initial value problem definition
Newton's laws of motion statement
Differentiable on an open set definition
Gradient definition
Higher partial derivatives definition
Picard–Lindelöf hypotheses definition
Iterated derivative definition
Picard operator definition
C^k function characterised by continuous derivatives statement
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Vector field definition
Picard–Lindelöf theorem statement
Conservative vector field definition
Classical mechanics definition
Momentum definition
Potential energy definition
Angular momentum definition
Central force definition
Newton's law of universal gravitation statement
N-body simulation definition

Tag	`R6VKPHSWM9`
URI	`m2://R6VKPHSWM9`
Kind	definition
Writing	W0 stub
MSC	85-10, 70F10, 65Z05
Defines	N-body simulation
Uses	n-body problem, Newton's law of universal gravitation, Ordinary differential equation
Changed	3 November 2026

N-body simulation

Definition

Remarks

Reading path

Further reading

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