| Abstract: | The connection between geometry and dynamics is a canonical subject of analytical mechanics. A very traditional issue of this topic is the transformation of the mechanical problem at hand into a shortest-path problem. This means the mathematical translation of the dynamical problem into a problem of finding the geodesic of a certain space. In the classical domain of conservative systems, especially following the famous book of Lanczos, this translating bridge is established by the usual condition of constant total energy. By nature, the motion of a particle with position-dependent mass is not a conservative problem. Therefore, the classical geometrical theory of mechanics is not straightforwardly applicable. Given that, we here aim at developing the geometrical theory for the mechanics of a position-dependent mass particle. This is our intended contribution. To our best knowledge, the content of our single investigation is original within this variable mass context. Our theory will be developed in the light of the inverse problem of Lagrangian mechanics, which will accordingly sets the variational framework. From that, we will demonstrate the proper generalization of Euler-Maupertuis’ principle and the following generalization of Jacobi’s principle, which, analogously to the classical procedure, can be seen as intermediate steps to enter geometrical arguments. Then, the corresponding geodesic will appear. Finally, as a closing result, a theorem on the mathematical equivalence between such geodesic and the equation of motion of a position-dependent mass particle will be proved. Our investigation aims at providing the reader with a fundamental contribution to the geometry of variable mass mechanics. |
