Symmetry aspects of nonholonomic field theories

TL;DR

This paper extends the geometric framework of nonholonomic field theories to include symmetries with horizontal components, deriving new equations and conservation laws relevant to practical models like the Cosserat rod.

Contribution

It introduces a new form of field equations for nonholonomic theories with general symmetries and develops a momentum equation framework for these symmetries, including generalized cases.

Findings

Derived a new form of nonholonomic field equations.

Established a momentum equation for nonholonomic symmetries.

Recovered energy conservation and identified a modified conservation law.

Abstract

The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, describing the evolution of the associated component of the momentum map. Keeping up with the underlying geometric philosophy, a small modification of the derivation of the momentum lemma allows us to treat also generalized nonholonomic symmetries, which are vector fields along a projection. Such symmetries arise for example in practical examples of nonholonomic field theories such as the Cosserat rod, for which we recover both energy conservation (a previously known result), as well as a modified conservation law associated with spatial translations.