Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Action Minimizing Solutions of The One-Dimensional $N$-Body Problem With
  Equal Masses   When we use variational methods to study the Newtonian $N$-body problem, the
main problem is how to avoid collisions. C.Marchal got a remarkable result,
that is, a path minimizing the Lagrangian action functional between two given
configurations is always a true (collision-free) solution, so long as the
dimension $d$ of physical space $\mathbb{R}^d$ satisfies $d\geq2$. But
Marchal's idea can't apply to the case of the one-dimensional physical space.
In this paper, we will study the fixed-ends problem for the one-dimensional
Newtonian $N$-body problem with equal masses to supplement Marchal's result.
More precisely, we first get the isolated property of collision moments for a
path minimizing the action functional between two given configurations, then,
if the particles at two endpoints have the same order, the path minimizing the
action functional is always a true (collision-free) solution; otherwise,
although there must be collisions for any path, we can prove that there are at
most $N! - 1$ collisions for any action minimizing path.

32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an