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

Action-minimizing solutions of the one-dimensional N-body 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 $${\mathbb {R}}^d$$ R d satisfies $$d\ge 2$$ 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 $$N! - 1$$ N ! - 1 collisions for any action-minimizing path. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Celestial Mechanics and Dynamical Astronomy Springer Journals

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

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Publisher
Springer Netherlands
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Physics; Astrophysics and Astroparticles; Dynamical Systems and Ergodic Theory; Aerospace Technology and Astronautics; Geophysics/Geodesy; Classical Mechanics
ISSN
0923-2958
eISSN
1572-9478
D.O.I.
10.1007/s10569-018-9830-z
Publisher site
See Article on Publisher Site

Abstract

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 $${\mathbb {R}}^d$$ R d satisfies $$d\ge 2$$ 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 $$N! - 1$$ N ! - 1 collisions for any action-minimizing path.

Journal

Celestial Mechanics and Dynamical AstronomySpringer Journals

Published: May 3, 2018

References

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