A Methodology for Finding Invariants of Motion for Asymmetric Systems with Gauge-Transformed Umbra Lagrangian Generated by Bond Graphs

The purpose of this article is to obtain conservation laws (invariants of motion) for different energy domains through the extended Noether theorem and bond graphs. Bond graphs are profitably used in representing the physics of a system as well as obtaining its umbra-Lagrangian. The article extends Lagrangian-Hamiltonian mechanics to deal with asymmetries in the system, which incorporates dissipative and nonpotential fields in a compact Lagrangian form, such that one may obtain invariants of motion through extension of Noether’s theorem. A detailed methodology is outlined in this article for obtaining the invariants of motion for a general class of asymmetric systems with a gauge-transformed umbra-Lagrangian. Symmetrization of an asymmetric system is introduced through the concept of gauge functions, for which the classical Noether theorem is extended over vector fields in the extended manifold comprising real and umbra displacements and velocities, as well as real time. A generalization of the variational principle or least action principle is also presented, which leads to the proposed form of the umbra-Lagrange equation through recursive minimization of functionals. Several illustrative examples are given to elucidate this concept in different physical contexts.