Conformal Killing vector fields and a virial theorem

TL;DR

This paper develops a virial theorem framework for Lagrangian systems on Riemann manifolds, focusing on affine virial functions linked to various types of conformal Killing vector fields, and establishes related theorems.

Contribution

It introduces a generalized virial theorem for systems on Riemann manifolds involving conformal Killing vector fields, extending classical results to more geometric contexts.

Findings

Virial theorems are established for Killing, homothetic, and conformal Killing vector fields.

The paper formulates the virial theorem both intrinsically and in local coordinates.

Affine virial functions associated with these vector fields are analyzed.

Abstract

The virial theorem is formulated both intrinsically and in local coordinates for a Lagrangian system of mechanical type on a Riemann manifold. An import case studied in this paper is that of an affine virial function associated to a vector field on the configuration manifold. The special cases of a virial function associated to a Killing, a homothetic and a conformal Killing vector field are considered and the corresponding virial theorems are established for this type of functions.