Action-minimizing solutions of the one-dimensional N-body problem

作者：Yu, Xiang^*; Zhang, Shiqing

来源：Celestial Mechanics and Dynamical Astronomy, 2018, 130(5): 37.

DOI：10.1007/s10569-018-9830-z

• N-body problem
• Collisions
• Variational methods
• Central configurations
• The fixed-ends problem

We supplement the following result of C. Marchal on the Newtonian N-body problem: A path minimizing the Lagrangian action functional between two given configurations is always a true (collision-free) solution when the dimension d of the physical space R-d satisfies d >= 2. The focus of this paper is on the fixed-ends problem for the one-dimensional Newtonian N-body problem. We prove that a path minimizing the action functional in the set of paths joining two given configurations and having all the time the same order is always a true (collision-free) solution. Considering the one-dimensional N-body problem with equal masses, we prove that (i) collision instants are isolated for a path minimizing the action functional between two given configurations, (ii) if the particles at two endpoints have the same order, then the path minimizing the action functional is always a true (collision-free) solution and (iii) when the particles at two endpoints have different order, although there must be collisions for any path, we can prove that there are at most N! - 1 collisions for any action-minimizing path.

• 出版日期2018-5
• 单位四川大学; 西南财经大学