Augmented Variational Principles and Relative Conservation Laws in Classical Field Theory

TL;DR

This paper introduces augmented variational principles to define covariant relative conservation laws, enabling measurement of energy differences between configurations in classical field theory, addressing observer dependence and divergence ambiguities.

Contribution

It proposes a novel covariant framework for relative conservation laws using augmented variational principles, linking divergence ambiguities and recent formalisms in the literature.

Findings

Provides a covariant method for defining relative conserved quantities.

Addresses observer dependence in conserved quantities.

Includes detailed examples from classical mechanics.

Abstract

Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities which measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental Physics are considered in detail, starting from classical Mechanics.