Is it possible to formulate least action principle for dissipative systems?

Abstract

A longstanding open question in classical mechanics is to formulate the least action principle for dissipative systems. In this work, we give a general formulation of this principle by considering a whole conservative system including the damped moving body and its environment receiving the dissipated energy. This composite system has the conservative Hamiltonian $H=K_1+V_1+H_2$ where $K_1$ is the kinetic energy of the moving body, $V_1$ its potential energy and $H_2$ the energy of the environment. The Lagrangian can be derived by using the usual Legendre transformation $L=2K_1+2K_2-H$ where $K_2$ is the total kinetic energy of the environment. An equivalent expression of this Lagrangian is $L=K_1-V_1-E_d$ where $E_d$ is the energy dissipated by the friction from the moving body into the environment from the beginning of the motion. The usual variation calculus of least action leads to the correct equation of the damped motion. We also show that this general formulation is a natural consequence of the virtual work principle.