Symmetry aspects of nonholonomic field theories

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 also treat 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) and a modified conservation law associated with spatial translations.

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

“…This is a meeting topic between honholonomic mechanics and classical field theories. The construction of the Lagrangian and Hamiltonian formalism, as well as other problems such as the study of symmetries and reduction have been analyzed for the k-symplectic formulation [72] and for the multisymplectic models in several works [10,95,96,97,98].…”

Tulczyjew's derivations and intrinsic field equations in classical field theories

“…The nonholonomic Cosserat rod [21,25,26] is an example of a nonholonomic field theory which describes the motion of a rod which is constrained to roll without sliding on a horizontal surface. A Cosserat rod can be thought of as a long and thin deformable body.…”

A historical review on nonholomic mechanics

“…The weakness of the theory is the lack of examples, which should be a priority in the future research (some recent work has been developed by Vankerschaver [135][136][137][138]). …”