A geometric approach to a generalized virial theorem

TL;DR

This paper presents a unified geometric framework for deriving virial theorems across classical and quantum mechanics using symplectic formalism, including systems with position-dependent mass.

Contribution

It introduces a generalized approach to virial theorems via symplectic formalism, applicable to both Hamiltonian and Lagrangian systems, and extends to quantum mechanics.

Findings

Derived virial theorems using symplectic formalism for classical systems.

Extended the approach to quantum systems, establishing a quantum virial theorem.

Analyzed systems with position-dependent mass within this geometric framework.

Abstract

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.