Generalized virial theorem for contact Hamiltonian systems

Abstract

We formulate and study a generalized virial theorem for contact Hamiltonian systems. Such systems describe mechanical systems in the presence of simple dissipative forces such as Rayleigh friction, or the vertical motion of a particle falling in a fluid (quadratic drag) under the action of constant gravity. We find a generalized virial theorem for contact Hamiltonian systems which is distinct from that obtained earlier for the symplectic case. The 'contact' generalized virial theorem is shown to reduce to the earlier result on symplectic manifolds as a special case. Various examples of dissipative mechanical systems are discussed. We also formulate a generalized virial theorem in the contact Lagrangian framework.

Quantum dissipation and the virial theorem

“…where k = mω 2 0 and µ = mγ. Eqn (12) is the same as that obtained via a generalized virial theorem in the contact Hamiltonian framework in [19]. As we shall show in the next subsection, this result [eqn (11)] differs significantly from that for a Brownian particle where the equation of motion contains random force (noise terms).…”

“…If G remains bounded in its time evolution, then one gets the simple result {G, H} = 0 (7) where angled brackets • denote time-averaging. Eqn (7) has been referred to as a hypervirial theorem, or simply a generalized virial theorem [6] (see also [17][18][19]). For a typical conservative mechanical system, one has H(q i , p i ) = K(p i ) + V (q i ) where K = i p 2 i /2m i and V are the kinetic and potential energies respectively.…”

“…This agrees with eqn (1) obtained by Clausius. Although, eqn (7) only describes conservative systems, it has been extended to the case of contact Hamiltonian dynamics, which describe simple dissipative systems in [19]. Let us consider two simple dissipative systems below.…”