CSC/ECE 506 Spring 2011/ch4a zz

Introduction

Area of the Cascadia subduction zone, including the Cascade Volcanic Arc (red triangles). The Garibaldi Volcanic Belt is shown here as three red triangles at the northernmost end of the arc.

The N-body problem stated as follows: Select the position and velocity of n celestial bodies as states. Given the initial condition of of N bodies, compute their states at arbitrary time T. Normally a three-dimensional space is considered for N-body problem. There is a simplified N-body problem called restricted N-body problem where the mass of some of the bodies is negligible. Several collection of remarkable restricted three-body motions can be found in [1].

Many mathematicians have proofed that it is impossible to find a general solution for n-body problem analytically[2]. The system could become unstable very easily. However, the problem can be solved numerically. The most common approach is to iterate over a sequence of small time steps. Within each time step, the acceleration on a body is approximated by the transient acceleration in the pervious time step. The transient acceleration on a single body can be directly computed by summing the gravity from each of the other N-1 bodies. While this method is conceptually simple and is the algorithm of choice for many applications, its O(N²)

The simulation of N-body system can be used from simulation of celestial bodies (gravitational interaction)to interactions of a set of particles (electromagnetic interaction).

References

[1] [1] Collection of remarkable three-body motions

[2] Diacu, F (01/01/1996). "The solution of the n-body problem". The Mathematical intelligencer (0343-6993), 18 (3), p. 66.