A generalization of the Helmholtz conditions for the existence of a first‐order Lagrangian using nonholonomic velocities

Abstract

AbstractThe paper presents a generalization of the Helmholtz conditions for the existence of a first‐order kinetic potential (Lagrangian) in that cases where the motion equations of mechanical, electrical, or electromechanical systems are given in terms of nonholonomic velocities. It is assumed that the transformation between holonomic and nonholonomic velocities is known. It is shown how the generalized Helmholtz conditions for the simultaneous existence of a first‐order Lagrangian and a first‐order dissipation function follow from the generalized Helmholtz conditions for systems with holonomic velocities. Moreover, the classical Helmholtz conditions in case of nondissipative systems can be directly generalized to motion equations which include nonholonomic velocities. In both cases the Boltzmann‐Hamel equations are used. The approaches are demonstrated by two examples: a heavy gyroscope and a 3‐dimensional rotating electrical machine.