Removing Collision Singularities from Action Minimizers for the N-body Problem with Free Boundaries
نویسندگان
چکیده
For the planar and spatial N-body problems, it has been proved by Marchal and Chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions. This important result has been extended by Ferrario-Terracini to Newtonian-type problems and equivariant problems, and has been used to construct many symmetric solutions for the Nbody problem. In this paper we are interested in action minimizing solutions in function spaces with free boundaries. Imposed to the function spaces are boundary conditions such that every mass point starts and ends on two transversal proper subspaces of R, d ≥ 2. We will prove that solutions for this minimizing problem with free boundaries are always free from collisions, including boundary collisions. This result can be used to construct certain classes of relative periodic solutions for the N-body problem.
منابع مشابه
Symmetry groups and non-planar collisionless action-minimizing solutions of the three-body problem in three-dimensional space
Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem in the three-dimensional space. First we give a finite complete list of symmetry groups fit to the...
متن کاملPeriodic Solutions of the Planar N-center Problem with Topological Constraints
In the planar N-center problem, for a non-trivial free homotopy class of the configuration space satisfying certain mild condition, we show that there is at least one collision free T -periodic solution for any positive T. We use the direct method of calculus of variations and the main difficulty is to show that minimizers under certain topological constraints are free of collision.
متن کاملOn the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem
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 Λ for the n-body problem (with potential of homogeneous degree −α, w...
متن کاملON THE SINGULARITIES OF THE CURVED n-BODY PROBLEM
We study singularities of the n-body problem in spaces of constant curvature and generalize certain results due to Painlevé, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection—a property that offers a new method of approaching the problem in this particular case.
متن کاملPlatonic Polyhedra , Topological Constraints and Periodic Solutions of the Classical N - Body Problem
We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N–body problem which, up to a relabeling of the N particles, are invariant under the rotation group R of one of the five Platonic polyhedra. The number N coincides with the order |R| of R and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangean action ...
متن کامل