A variational characterization of the Lagrangian solutions of the three-body problem (Q2738873)

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 \textit{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 \(k_{ij}\neq 0\) 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).