Inverse Problem for Lagrangian Dynamics for Multi-Degree-of-Freedom Systems with Linear Damping

This paper deals with the inverse problem for Lagrangian dynamics for linear multi-degree-of-freedom systems. New results for linearly damped systems are obtained using extensions of results for single-degree-of-freedom systems. First, for a two-degree-of-freedom linear system with linear damping, the conditions for the existence of a Lagrangian are explicitly obtained by solving the Helmholtz conditions. Next, since the Helmholtz conditions are near-impossible to solve for general n-degree-of-freedom systems, a new simple procedure that does not require the use of the Helmholtz conditions and that is easily extended to n-degree-of-freedom linear systems, is developed. The emphasis is on obtaining the Lagrangians for these multi-degree-of-freedom systems in a simple manner, using insights obtained from our understanding of the inverse problem for single- and two-degree-of-freedom systems. Specifically we include systems that commonly arise in linear vibration theory with positive definite mass matrices, and symmetric stiffness and damping matrices. This method yields several new Lagrangians for linear multi-degree-of-freedom systems. Finally, conservation laws for these damped multi degree-of-freedom systems are found using the Lagrangians obtained.