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.

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.

A generalization of the virial theorem to nonholonomic Lagrangian systems is given. We will first establish the theorem in terms of Lagrange multipliers and later on in terms of the nonholonomic bracket.

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