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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 $k=\left({k}_{12},{k}_{23},{k}_{31}\right)$ 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 $k=\left(1,1,1\right)$ or $k=\left(-1,-1,-1\right)$ (Theorem 1), whilst the homothetic orbits are the minimizers of the action in the case ${k}_{ij}\ne 0$ for all $\left(i,j\right)$ (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:
 70F07 Three-body problems