On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem

Abstract

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space for the n-body problem (with potential of homogeneous degree alpha, with alpha>0) which ensures that the restriction of the Lagrangian action to the space of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. Many of the already known periodic orbits can be proved to exist by this result, and several new orbits are found with some appropriate choice of G.

58 pages, 10 figures. Second revised version

Cited by in corpus (27)

• Positive periodic solutions of singular systems
• Action minimizing orbits in the n-body problem with simple choreography constraint
• Trace formula for linear Hamiltonian systems with its applications to elliptic Lagrangian solutions
• Globally minimizing parabolic motions in the Newtonian N-body problem
• Unchained polygons and the N-body problem
• Spatial double choreographies of the Newtonian $2n$-body problem
• Entire minimal parabolic trajectories: the planar anisotropic Kepler problem
• On the dihedral n-body problem
• On weak KAM theory for N-body problems
• Existence of symmetric central configurations
• Variational proof of the existence of the super-eight orbit in the four-body problem
• A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem
• On the free time minimizers of the Newtonian N-body problem
• Classification of symmetry groups for planar n-body choreographies
• Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem
• New Periodic Solutions for Newtonian $n$-Body Problems with Dihedral Group Symmetry and Topological Constraints
• Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem
• Five-body choreography on the algebraic lemniscate is a potential motion
• On the stability of periodic N-body motions with the symmetry of Platonic polyhedra
• Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem
• Choreographies in the Discrete Nonlinear Schrödinger Equations
• Braids of the N-body problem by cabling a body in a central configuration
• Geodesic rays of the $N$-body problem
• Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems
• Stability of Periodic Orbits by Conley-Zehnder index theory
• New solutions for the symmetrical n-body problem through variational approach and optimisation techniques
• Null Angular Momentum and Weak KAM Solutions of the Newtonian $N$-Body Problem