Virial theorem in quasi-coordinates and Lie algebroid formalism

Abstract

In this paper, the geometric approach to the virial theorem developed in [1] is written in terms of quasi-velocities (see [2]). A generalization of the virial theorem for mechanical systems on Lie algebroids is also given, using the geometric tools of Lagrangian and Hamiltonian mechanics on the prolongation of the Lie algebroid.

Conformal Killing vector fields and a virial theorem

“…As virial-like relations can be directly established in terms of the Lagrangian function and are not so easily derivable in the Hamiltonian formalism we will mainly restrict ourselves to the Lagrangian formalism, even if the final expressions can be translated to the Hamiltonian language. The extension of some of such results to the framework of mechanics in Lie algebroids was developed in [9]. This paper tries to develop analogous results in the particular case of mechanical type Lagrangians, and in this case conformal Killing vector fields will be shown to play a very relevant role.…”

Generalized virial theorem for contact Hamiltonian systems

“…There is an analogous Darboux's theorem for contact manifolds which says that near any point, it is possible to define local (Darboux) coordinates (s, q i , p i ) such that η = ds − p i dq i (10) and thus ξ = ∂/∂s in local coordinates. Let us note that since locally dη = dq i ∧ dp i , one may think of a contact manifold to be locally of the form M c = M s × R where M s is a (exact) symplectic submanifold (codimension one) of M c such that θ = Φ * η is the symplectic potential on M s and Φ : M s → M c is the inclusion map.…”

“…A generalized version of the virial theorem, called the 'hypervirial' theorem was studied in [6] and certain hypervirial relations have been recently obtained in [7]. The formulation of the hypervirial theorem or simply the generalized virial theorem was studied in [8][9][10][11] using methods of symplectic geometry, wherein various mechanical examples including position-dependent mass systems were discussed. The purpose of this note is to study the generalized virial theorem for contact Hamiltonian systems which describe some simple dissipative systems in mechanics.…”

Generalized virial theorem for the Liénard-type systems

“…In this paper we consider geometrical mechanics approach and necessity of geometric treatment of VT [5,6] can be argued as follows. The standard VT is based on the transformation properties of kinetic and potential energies under dilations, and therefore is only valid for systems with R n as configuration space.…”