Revisiting the generalized virial theorem and its applications from the perspective of contact and cosymplectic geometry

Abstract

The generalization of the virial theorem, introduced by Clausius in statistical mechanics, has recently been carried out in the framework of geometric approaches to Hamiltonian and Lagrangian theories and it has been formulated in an intrinsic way. It is here revisited not only in its more general situation of a generic vector field but mainly from the perspective of contact and cosymplectic geometry. The previous generalizations allowing virial like relations from one-parameter groups of non-strictly canonical transformations and the rôle of Killing and conformal Killing vector fields for Lagrangians of a mechanical type are here completed with the theory for contact Hamiltonian, as well as gradient and evolution, vector fields. The corresponding theories in the framework of cosymplectic geometry and the particular case of time-dependent vector fields are also developed.