Classical Solutions for a Perturbed N-Body System

Antonio, Gianfausto Dell'

doi:10.1007/978-1-4612-4126-3_1

Cited by 4 publications

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“…In particular, a system of point-particles interacting through Newtonian-like potential (the Newtonian-like N -body problem (NBP)) has been extensively studied. One can refer to the monograph [1] and to the long paper [4]. The aim of variational methods is to seek the periodic solutions of the equation of motion as critical points of the action functional of the system: the action is the integral of the Lagrangian associated to the system, the resulting functional is then studied on a suitable subset of the continuous closed trajectories on the configuration space of the system.…”

Section: Introductionmentioning

confidence: 99%

The topology of the planar three-body problem with zero total angular momentum and the existence of periodic orbits

Sbano¹

1998

Nonlinearity

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We consider the planar three-body problem (3BP) with a Newtonian-like potential of the form i =j m i m j x i − x j −α with α 2 and also the Newtonian case α = 1. We study the least action principle on the manifold defined by the vanishing of the total angular momentum. Using the topology of the reduced configuration space we are able to find a class of trajectories on which a generalization of the Poincaré inequality holds, this then enables us to apply the direct method of calculus of variations. We prove the existence of a periodic solution: for the Newtonian potential (α = 1), this periodic solution has at most a finite number of collisions. For α 2 the solution is in the homotopic class of the trajectories that go through at least three different collinear configurations. This result is a direct consequence of the first homotopy group of the reduced configuration space without coincidence set, this group is isomorphic to the free-generated group Z * Z.

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Section: Introductionmentioning

confidence: 99%

The topology of the planar three-body problem with zero total angular momentum and the existence of periodic orbits

Sbano¹

1998

Nonlinearity

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Geometric characterizations for variational minimization solutions of the 3-body problem

Zhang

1999

Chin.Sci.Bull.

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For any given positive masses it is proved that the variational minimization solutions of the 3-body problem in ii3 or A? are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in IFF or ?t2 are also planar equilateral triangle circular solutions. Keywords : 3-body problem, d t i o n a l minimhatinn, equilateral triangle drcnlar solution.THE classical N-body problem in the celestial mechanics concerns the motion of N point masses governed by the universal gravitation law. The motion q = ( q , ( t ) , , qfi( t ) ) satisfies the following system:where qi E lRK with K = 3 or 2 being the position, mi is the mass of the i-th particle for i = 1 , ... , N respectively, and -U( g ) is the potential function given by and U,, denotes the gradient of U with respect to q i . By K > 0 we denote the universal gravitation constant. After Rabinowitz' pioneering work['] in 1978. a great amount of contributions to periodic solutions of the N-body problems as well as restricted N-body problems has been made via variational minimization method on r-periodic function spaces. Two beautiful survey works are refs. [ 2 1 and [ 3 1 ( cf. also refs [4-81 and references therein) . Many mathematicians work on r/2-antiperiodic function paces, where a function q = q ( t ) is called r/2-antiperiodic, if q ( t + r / 2 ) = -q ( t ) for all t E Z . Via the variational minimization methods, the existence of non-collision periodic solutions was proved for the 2-body problem by Degiovanni and Giannoni in ref. [ 7 ] of 1989, and for the 3-body problem in Yt3 by Serra and Terracini in ref. [8] of 1992. Note that ref. [ 81 depends on the famous and heavy work of Sundman at the begrnning of this century[91 and does not apply to the 3-body problem in 3'.

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Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

Zhang

2000

Acta Math Sinica

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Classical Solutions for a Perturbed N-Body System

Cited by 4 publications

References 31 publications

The topology of the planar three-body problem with zero total angular momentum and the existence of periodic orbits

The topology of the planar three-body problem with zero total angular momentum and the existence of periodic orbits

Geometric characterizations for variational minimization solutions of the 3-body problem

Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

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