Conservation laws for a gauge-variant umbra-Lagrangian in classical mechanics using bond graphs

Abstract

In this article, conservation laws (invariants of motion) have been derived from symmetry operations for different energy domains. The formulation has been derived through an extended Noether’s theorem and the bond graphs. An additional time-like variable has been used which is termed an umbra and this notation was appended to all kinds of energies and the Lagrangian itself. We extend Lagrangian—Hamiltonian mechanics to deal with asymmetries in the system, which incorporates dissipative and non-potential fields. The gauge-functions have been used to symmetrize the umbra-Lagrangian of the system. Several new examples have been incorporated to elucidate the concept. The first example presents the complex gauge functions for a radiation thermal system, where a mass of ice is heated by the radiation process. The conservation law for such a system is obtained. In the second example, gauge functions for the system with time-varying stiffness are considered, which clearly expresses the usefulness of the umbra-Lagrangian method. In another example, gauge functions for a control system have been obtained, which shows the applicability of the method in different energy domains.