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.

Kirillov structures and reduction of Hamiltonian systems by scaling and standard symmetries

“…Note however, that this 𝑝-projection of 𝑋 {⋅,⋅} 𝔤 * 𝜉 𝓁 is just the vector field 𝑋 ℎ 𝜉 ∈ 𝔛(ℙ𝔤 * ), which is locally characterized by (22).…”

“…• The final reduced dynamics is the vector field 𝑋 ℎ 𝜉 on ℙ𝔤 * described in (49), which is the symbol of [⋅, ℎ 𝐺 𝜉 ] 𝐿 * and whose local expression is (22).…”