An Elemental Overview of the Nonholonomic Noether Theorem

Abstract

Noether theorem plays a central role in linking symmetries and first integrals in Lagrangian mechanics. The situation is different in the nonholonomic context, but in the last decades there have been several extensions of Noether theorem to the nonholonomic setting. We provide an overview of this subject which is as elementary as possible.

Conservation of energy and momenta in nonholonomic systems with affine constraints

“…The inspection of this function reveals that, for a given system, the set of all reaction forces exerted by the constraints on the constrained motions might be (and typically is) smaller than the set of all reaction forces that satisfy the condition of ideality. The reason is that for a given system the active forces that act on the system are fixed, while the notion of ideality makes reference to all possible active forces that might possibly act on the system (see [19] for a discussion of this fact). Consequently, any vector that annihilates the linear part of the constraint may be an ideal reaction force, but for a given system, the class of reaction forces actually exerted by the constraint may be a subset of this annihilator.…”

Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

“…This contributes to the recent efforts to understand the mechanisms responsible for the existence of first integrals that are linear in velocities in nonholonomic mechanics (see e.g. [24,11,12,1]).…”

Symmetry and conservation laws in non-holonomic mechanics

“…The equations of motion of a mechanical system subject to ideal kinetic constraints are well known. [1][2][3][4][5][9][10][11][12]24 In this subsection, we add a few remarks on the subject.…”