Slow Motion of Small Particles in the Fluid - A Variational Principle
In this paper we discuss the variational approach in the context of motion of aerosol particles in an atmospheric viscous fluid. Consider an aerosol particle treated as a macroscopic body moving in the atmosphere, where the presence of the friction phenomena results in the appearance of certain additional frictional forces. Can we describe the system in terms of a variational principle taking into account dissipation?
To answer this question we explore a modification of classical Hamiltonian action by making an assumption that the Lagrangian of the irreversible motion Lirr is an action dependent rather than time dependent quantity, so that action A or action variable x0 = -A should explicitly appear in Lirr. Since in this formulation Lirr does not contain explicitly time t, the irreversibility is no longer associated with the time dependent properties of the Lagrangian Lirr (as in many previous trials). Speaking in more concrete words, the proposed approach is accomplished by introducing an extra additive term into Lagrangian L. This assumption allows to secure the constancy of the autonomous Hamiltonian H, and has some other virtues that are discussed in the text. We shall also stress that with this approach one shall be are able to discuss the energy conservation and achieve a satisfactory set of equations of motion.