Conservation laws for dissipative systems possessing classical normal modes

Linear, viscously damped dynamical systems whose matrix coefficients satisfy a certain commutativity condition are known to exhibit the same normal modes as the ones associated with the same system in the absence of damping. Such dissipative systems are said to possess classical normal modes. In the present study, it is shown that the original equation of motion in the displacemnt vector which exhibits velocity coupling can be transformed into an equation where the velocity term is no longer present, but whose matrix coefficients are time-dependent. A commutativity condition further reduces the resulting equation into one with constant coefficients. Global and modal energy conservation laws are constructed for this equation governing the undamped position vector, and the results are expressed in terms of the original damped physical vector. As a by-product of the analysis, independent first integrals equal in number to the degrees of freedom of the system are obtained.