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A variational characterization of the Lagrangian solutions of the three-body problem. (Une caractérisation variationnelle des solutions de Lagrange du problème plan des trois corps.) (French) Zbl 1034.70007
The planar three-body problem is a mechanical system which consists of three mass points m 1 , m 2 , m 3 which attract each other according to the Newtonian law. A significant problem is the existence of its periodic solutions. Only a limited number of these solutions is known, including in particular the Lagrangian homographic and homothetic solutions. These ones are considered by the author. Following R. Montgomery [Nonlinearity 11, 363–376 (1998; Zbl 1076.70503)], a periodic motion of three masses can be characterized by the triplet of winding numbers k=(k 12 ,k 23 ,k 31 ) defining a particular homology class. Fixing k, the action functional can be restricted to the particular set of T-periodic loops. Studying the minimization problem on this set, the author proves that the homographic Lagrangian solutions correspond to the choice k=(1,1,1) or k=(-1,-1,-1) (Theorem 1), whilst the homothetic orbits are the minimizers of the action in the case k ij 0 for all (i,j) (Theorem 2). It is noticed that the idea to minimize the action on the particular homology class was already introduced by H. Poincaré (1896).
MSC:
70F07Three-body problems