Inverse approach in ordinary differential equations: applications to Lagrangian and Hamiltonian mechanics
The authors study inverse problems of ordinary differential equations which consist in finding a differential system satisfying a set of given properties. For instance, to find what are the differential systems in \(\mathbb{R}^N\) having a given set of invariant hypersurfaces, or having a set of first integrals. The first inverse problem was stated and solved by Newton determining the potential field of a force that ensures the planetary motion in accordance to Kepler's laws. In 1877, \textit{J. Bertrand} [C. R. Acad. Sci., Paris 84, 671--674 (1877; JFM 09.0638.01)] proved that the expression of the Newton's force of attraction can be obtained directly from Kepler's first law. Bertrand stated also a more general problem of determining a positional force, under which a particle describes a conic section under any initial conditions. These ideas were developed in series of works until our days. Hence, the origin of the inverse problem comes from celestial mechanics. According to the second Newton law, the mentioned inverse problems are equivalent to determine second-order differential equations from given properties that must be satisfied. The inverse problem of finding more general differential systems of first order satisfying a set of given properties was stated by \textit{N. P. Erugin} [Prikl. Mat. Mekh. 16, 659--670 (1952; Zbl 0047.32805)] for dimension 2 and later developed by \textit{A. S. Galiullin} [Inverse problems of dynamics. (Obratnye zadachi dinamiki) (Russian). Moskva: ''Nauka'' (1981; Zbl 0516.70002)]. In the present work, the authors propose a new approach to an inverse problem which uses the Nambu bracket [\textit{Y. Nambu}, Phys. Rev. D (3) 7, 2405--2412 (1973; Zbl 1027.70503)]. The authors deduce new properties of this bracket which play a very important role in the proof of all the results of this work and in their applications. The applications of the Nambu bracket given in this paper are original and present a new direction in order to develop Nambu's ideas. In the first part of the work, the authors characterize, under very general assumptions, the ordinary differential equations in \(\mathbb{R}^N\) which have a given set of either \(M\) partial integrals, or \(M < N\) first integrals, or \(M = N\) partial and first integrals. For such systems, the authors provide necessary and sufficient conditions for the integrability, i.e., for the existence of \(N - 1\) independent first integrals. In the rest of the work, the authors apply the results of the first part. Firstly, they state and solve the inverse problem for the constrained Lagrangian mechanics: For a given natural mechanical system with \(N\) degrees of freedom they determine the most general field of force depending only on the position of the system satisfying a given set of constraints which are linear in the velocity. One of the main objectives of this inverse problem is to study the behavior of the constrained Lagrangian systems with linear constraints with respect to the velocity in a way different from the classical approach deduced from the d'Alembert-Lagrange principle. Secondly, the authors study the inverse problem for constrained Hamiltonian systems. That is, for a given submanifold \(\mathcal{M}\) of a symplectic manifold \(\mathbb{M}\), the authors determine the differential systems having the submanifold \(\mathcal{M}\) invariant under their flows.
scientific article; zbMATH DE number 6380036
scientific article