The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions

Abstract

We consider nonholonomic systems with linear, time-independent constraints subject to positional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator distribution R degrees, the fibers of which are the annihilators of the set of all values taken by the reaction forces on the fibers of the constraint distribution. We show that this distribution, which can be effectively computed in specific cases, plays a central role in the study of first integrals linear in the velocities of this class of nonholonomic systems. In particular we prove that, if the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, then an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution R degrees. Since the fibers of R degrees contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved momenta than what was known so far. Some examples are given

Moving energies as first integrals of nonholonomic systems with affine constraints

“…, k). For details see [16]; in the case of linear constraints, these or analogue expressions are given in [1,2,14]. We note that the restriction of R to M is independent of the arbitrariness that affects the choices of the vector field Z, of the matrix S and of the vector s, see [16].…”

“…We need to introduce now the so-called reaction-annihilator distribution R • , from [14,16]. This object plays a central role in the conservation of energy and of moving energies of nonholonomic systems with affine constraints [16,17] (as well as in the conservation of momenta in nonholonomic systems with either linear or affine constraints [14,16]).…”

“…A geometric characterization of the vector fields whose momenta are first integrals of a nonholonomic system (L, Q, M) with affine constraints does not exist. However, just as in the case of systems with linear constraints, see Proposition 2 of [14], we may characterize those among them which have another property as well.…”

An Elemental Overview of the Nonholonomic Noether Theorem

“…Nevertheless, the annihilators R • q ⊂ T q Q of these sets are linear spaces and are thus the fibers of a distribution R • on Q, possibly of non-constant rank and non-smooth. Since the space R • q contains all tangent vectorsq ∈ T q Q which annihilate all possible values of the reaction forces on constraint motions through q, R • was called the reaction-annihilator distribution [22]. Clearly…”

“…For further analysis of the reaction-annihilator distribution, and some examples, see [22]. are the tangent spaces to the orbits of the action Ψ.)…”