A Systematic Illustration on Reduction of N-body Problem with Application to Molecular Systems

6n 3n T 6n ≅ and * 3 6 n T n ≅ . This means that 6N possibly-nonlinear first order ODEs altogether describe the dynamics. It will be great if the dimension could be reduced. In fact, such idea was initiated by pioneer physicists ~400 years ago in solving Kepler problem [1, 2], in which the 2-body problem of sunearth-gravity is studied with the aid of conservations of total linear momentum and total angular momentum. Now, people could follow a systematic way to rigidly carry out reductions. Momentum maps are introduced to depict the essence of conserved quantities including total linear momentum and total angular momentum, and N-body problems could be studied by symplectic reduction theory which utilizes symmetries [3, 4, 5]. In this article, a geometrical reduction of 3-body problem will be described, with both the zero angular momentum and the non-zero angular momentum cases discussed. What follows is a discussion on the singularity issue associated with the reduction of 2body problem. Then Yanao et al. [6]’s work of a further decrease of dimension in 6-body problem will be introduced and discussed. Applications and future directions will conclude the article.