A variational formulation for dissipative systems

It is well known that classical Lagrangians do not exist for dissipative systems. The paper presents three formulations of a mathematical Lagrangian for a dissipative system. The first one depends on the generalized coordinate x, its velocity x˙, the image variable y and its velocity y˙. This Lagrangian can be obtained directly from differential equations of a dynamic system. For linear reciprocal dissipative systems, the image variable can be substituted by the generalize coordinate with reversed time scales. The second formulation which can resolve the problem for nonlinear dynamical systems depends on kinetic energy, potential energy, and the time integral of a dissipation function. The last formulation which is a time dependent function can be obtained for a special type of linear system