The inverse problem of Lagrangian mechanics for a non-material volume

摘要

The appropriate consideration of non-material volumes at the level of analytical mechanics is an ongoing research field. In the present paper, we aim at demonstrating the principle of stationary action that is able to yield the proper form of Lagrange's equation in the context, namely the Lagrange's equation in the form derived by Irschik and Holl (Acta Mech 153(3-4):231-248, 2002). Such issue will here be interpreted as being the inverse problem of Lagrangian mechanics for a non-material volume. The classical method of Darboux (Le double dagger ons sur la Th,orie G,n,rale des Surfaces. Gauthier-Villars, Paris, 1891) will be used as the solution technique. This means that our discussion will be restricted to the case of a single degree of freedom. Having such principle of stationary action at hand, the corresponding Hamiltonian formalism will be written in accordance with the classical theory. Furthermore, a conservation law will be demonstrated for the time-independent case. At last, two simple examples will be addressed in order to illustrate the applicability of the proposed formulation. The reader may find some mathematical analogies between the upcoming content and that discussed by Casetta and Pesce (Acta Mech, 2013. doi:10.1007/s00707-013-1004-1) in considering the inverse problem of Lagrangian mechanics for Meshchersky's equation. The mathematical formulation which will be outlined in the present paper is thus expected to consistently situate non-material volumes within the classical variational approach of mechanics.