Action minimizing orbits in the n-body problem with simple choreography constraint

Abstract

In 1999 Chenciner and Montgomery found a remarkably simple choreographic motion for the planar 3-body problem (see \cite{CM}). In this solution 3 equal masses travel on a eight shaped planar curve; this orbit is obtained minimizing the action integral on the set of simple planar choreographies with some special symmetry constraints. In this work our aim is to study the problem of $n$ masses moving in $\RR^d$ under an attractive force generated by a potential of the kind $1/r^{\alpha }$, $\alpha >0$, with the only constraint to be a simple choreography: if $q_1(t),...,q_n(t)$ are the $n$ orbits then we impose the existence of $x \in H^1_{2 π}(\RR,\RR^d)$ such that $q_i(t)=x(t+(i-1) τ), i=1,...,n, t \in \RR, $ where $\tau =2\pi /n$. In this setting, we first prove that for every $d,n \in \NN$ and $\alpha >0$, the lagrangian action attains its absolute minimum on the planar circle. Next we deal with the problem in a rotating frame and we show a reacher phenomenology: indeed while for some values of the angular velocity minimizers are still circles, for others the minima of the action are not anymore rigid motions.

24 pages; 4 figures; submitted to Nonlinearity

References in corpus (1)

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

Cited by in corpus (12)

• Globally minimizing parabolic motions in the Newtonian N-body problem
• Unchained polygons and the N-body problem
• Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory
• Spatial double choreographies of the Newtonian $2n$-body problem
• A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem
• Variational proof of the existence of the super-eight orbit in the four-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
• Choreographies in the Discrete Nonlinear Schrödinger Equations
• New solutions for the symmetrical n-body problem through variational approach and optimisation techniques
• New Periodic Solutions for Some Planar $N+3$-Body Problems with Newtonian Potentials
• Continuous Choreographies as Limiting Solutions of $N$-body Type Problems with Weak Interaction