The inverse problem of Lagrangian mechanics for Meshchersky’s equation

Abstract

Variable-mass systems are not included in the conventional domain of the analytical and variational methods of classical mechanics. This is due to the fact that the fundamental principles of mechanics were primarily conceived for constant-mass systems. In the present article, an analytical and variational formulation for variable-mass systems will be proposed. This will be done from the solution of the here called ‘inverse problem of Lagrangian mechanics for Meshchersky’s equation’. The first problem of this nature was posed in 1887, by Helmholtz (J. reine angew. Math. 100:137–166, 1887). Investigations on the matter are far from being exhausted. Within mechanics, it means the construction of a Lagrangian from a given equation of motion. To the authors’ best knowledge, aiming at general results, the inverse problem of Lagrangian mechanics has not been properly connected to Meshchersky’s equation yet. This is the main goal of this article. We will address the issue by assuming that mass depends on generalized coordinate, generalized velocity and on time. After the construction of a Lagrangian from Meshchersky’s equation, a general and unifying mathematical formulation will emerge in accordance. Therefore, variable-mass systems will be accommodated at the level of analytical mechanics. A variational formulation, which will be written via a principle of stationary action, and a Hamiltonian formulation will be both stated. The latter could be read as the ‘Hamiltonization’ of variable-mass systems from the solution of the inverse problem of Lagrangian mechanics. An energy-like conservation law will naturally appear from the simplification of the general theory to the case of a system with mass solely dependent on a generalized coordinate.