Clausius/Cosserat/Maxwell/Weyl Equations: The Virial Theorem Revisited

TL;DR

This paper revisits the mathematical foundations of thermodynamics and gauge theory using advanced differential geometric methods, revealing deep connections between classical equations and the Spencer operator related to the conformal group.

Contribution

It demonstrates that Clausius, Cosserat, Maxwell, and Weyl equations are formal adjoints of the Spencer operator, linking them to the conformal group and extending the virial theorem with modern geometric tools.

Findings

Clausius/Cosserat/Maxwell/Weyl equations are adjoints of the Spencer operator.

The virial theorem is justified and extended using differential geometry.

Group actions fundamentally underpin the classical equations studied.

Abstract

In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincar{\'e} introduced a duality principle in analytical mechanics in order to study lagrangians invariant under the action of a Lie group of transformations. In 1909, the brothers E. and F. Cosserat discovered another approach for studying the same problem though using quite different equations. In 1916, H. Weyl considered again the same problem for the conformal group of transformations, obtaining at the same time the Maxwell equations and an additional specific equation also involving the trace of the impulsion-energy tensor. Finally, having in mind the space-time formulation of electromagnetism and the Maurer-Cartan equations for Lie groups, gauge theory has been created by C.N. Yang and R.L. Mills in 1954 as a way to introduce in physics the differential geometric methods available at that time, independently of any group action, contrary to all the previous approaches. The main purpose of this paper is to revisit the mathematical foundations of thermodynamics and gauge theory by using new differential geometric methods coming from the formal theory of systems of partial differential equations and Lie pseudogroups, mostly developped by D.C Spencer and coworkers around 1970. In particular, we justify and extend the virial theorem, showing that the Clausius/Cosserat/Maxwell/Weyl equations are nothing else but the formal adjoint of the Spencer operator appearing in the canonical Spencer sequence for the conformal group of space-time and are thus totally dependent on the group action. The duality principle also appeals to the formal adjoint of a linear differential operator used in differential geometry and to the extension modules used in homological algebra.