Lagrangians and Integrals of Motion for Multi-Degree-of-Freedom Linear Systems with Gyroscopic and Circulatory Forces

This paper deals with multi-degree-of-freedom linear dynamic systems that may be subjected to damping, gyroscopic, and circulatory forces. Under suitable conditions, Lagrangians are obtained for such systems. These new results include and significantly generalize previous work reported on Lagrangians and invariants of motion for multi-degree-of-freedom linear classically damped systems in that they permit the presence of gyroscopic and circulatory terms when modeling physical systems. To delineate the compass of applicability of these results, the conditions under which the presence of such terms can be included in the dynamic description are provided. An invariant of the motion, or conservation law, is also obtained for such general systems. The invariant is shown to be a natural generalization of the well-known conservation-of-energy principle that is applicable to undamped multi degree-of-freedom potential systems.