  The virial theorem, introduced by Clausius in statistical mechanics, and
later applied in both classical mechanics and quantum mechanics, is studied by
making use of symplectic formalism as an approach in the case of both the
Hamiltonian and Lagrangian systems. The possibility of establishing virial's
like theorems from one-parameter groups of non-strictly canonical
transformations is analysed; and the case of systems with a position dependent
mass is also discussed. Using the modern symplectic approach to quantum
mechanics we arrive at the quantum virial theorem in full analogy with the
classical case.
