[Paper Review] Periodic Solutions for $N$-Body-Type Problems

Fengying Li, Shiqing Zhang|arXiv (Cornell University)|Aug 13, 2014

Spacecraft Dynamics and Control13 references3 citations

TL;DR

This paper establishes the existence of unbounded sequences of non-collision periodic solutions for N-body-type problems with strong force potentials at the origin and sub-quadratic growth at infinity. Using Ljusternik-Schnirelmann theory and a relaxed growth condition on the potentials, the authors prove the existence of infinitely many critical values for the Lagrangian action functional, extending prior results by weakening the non-positivity assumption on the potential while preserving the variational structure.

ABSTRACT

We consider non-autonomous $N$-body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity, and using Ljusternik-Schnirelmann theory, we prove the existence of unbounded sequences of critical values for the Lagrangian action corresponding to non-collision periodic solutions.

Motivation & Objective

• To extend the existence of infinitely many non-collision periodic solutions for N-body-type problems beyond previous assumptions.
• To relax the condition that potentials must be non-positive (as in earlier works) while preserving the existence of unbounded sequences of critical values.
• To establish the validity of Ljusternik-Schnirelmann theory under a sub-quadratic growth condition on the potential, allowing for stronger singularities at the origin.
• To prove the local Palais-Smale condition for the Lagrangian action functional under the new growth assumption, enabling the application of abstract critical point theory.

Proposed method

• The authors use Ljusternik-Schnirelmann theory to analyze the critical points of the Lagrangian action functional associated with the N-body system.
• They introduce a relaxed growth condition (V4') that bounds the potential from above by a sub-quadratic power of the relative distance, allowing for positive values at large distances.
• The proof relies on verifying the local Palais-Smale condition by establishing uniform boundedness of the kinetic energy in a minimizing sequence via Wirtinger’s inequality and case analysis on the size of relative positions.
• The category theory argument is applied by showing that the sublevel sets of the action functional have finite category, and the functional tends to infinity near the boundary of the domain.
• A symmetric structure is exploited: the solution paths satisfy $ x^k(t + T/2) = -x^k(t) $, which implies zero mean and allows the use of Wirtinger’s inequality to control the $ L^2 $-norm of velocity.
• The proof combines estimates on the potential energy using the growth condition and the kinetic energy using spectral inequalities, leading to uniform boundedness of the sequence in the Sobolev space $ H^1 $.

Experimental results

Research questions

• RQ1 Can the existence of infinitely many non-collision periodic solutions be established under a weaker non-positivity assumption on the N-body potential?
• RQ2 Does the Lagrangian action functional for N-body problems with strong force singularities and sub-quadratic growth admit an unbounded sequence of critical values?
• RQ3 Can the Palais-Smale condition be verified under a relaxed growth condition that allows positive potential values at large distances?
• RQ4 Is Ljusternik-Schnirelmann theory applicable to N-body problems when the potential is not necessarily non-positive but satisfies a sub-quadratic upper bound?
• RQ5 What structural assumptions on the potential and solution symmetry are sufficient to ensure compactness in the variational setting?

Key findings

• The paper proves the existence of an unbounded sequence of critical values for the Lagrangian action functional under the new condition (V4'), which relaxes the requirement that $ V_{ij} \leq 0 $.
• The authors establish the local Palais-Schnirelmann condition by showing that any sequence with bounded action has a weakly convergent subsequence, with uniform bounds on kinetic energy derived via Wirtinger’s inequality.
• The category of the domain is infinite, a key requirement for applying Ljusternik-Schnirelmann theory, and this is confirmed using symmetric structure and known category estimates.
• The proof handles three cases based on the size of relative positions: all small, all large, and mixed; in each case, the action remains bounded below, ensuring uniform control.
• The boundedness of the kinetic energy is shown via a case analysis: when all relative distances are large, the sub-quadratic growth ensures control; when some are small, the potential is uniformly bounded below.
• The final conclusion is that the Lagrangian action functional has infinitely many critical points corresponding to non-collision T-periodic solutions, even when the potential is not non-positive, provided the growth condition (V4') holds.