Noether Theorem, 90 years on

Elastica as a dynamical system

“…There is a vast amount of work on symmetry principles and conservation laws following Noether's fundamental paper [17]. Three works in particular are noteworthy: the monographs by Logan [16] and Kosmann-Schwarzbach [9], and a review by Krupkova [11].…”

Uniform motions in central fields

“…Recall that a vector field Ξ ∈ C is the constraint Noetherian symmetry of the nonholonomic constrained system [α Q ] if L Ξ ι * θ λ is a constraint 1-form [9] (L Ξ denotes the Lie derivative along a vector field Ξ).…”

“…The constraint version of the Noether's theorem is formulated in [9]: Let λ be a Lagrangian on J 1 Y and Ξ be a constraint symmetry of the nonholonomic constrained system [α Q ]. Then along every constrained pathγ the following identities are satisfied,…”

“…The word isotachytonic was introduced in [21] from Greek; iso = uniform; τ αχύτ ητ α = velocity. Since isotachytonic constraint represents the nonholonomic constraint nonlinear with respect to components of the velocity, it is necessary to adopt a modern approach based on the geometric concept of nonholonomic mechanical systems [6,7,9,11,13]. This problem posses some interesting aspects concerning proper kinematics (specific trajectories), understanding the role of the Chetaev constrained force and its active contribution to the energetic balance of the system, understanding of the constraint Noetherian conservation laws as a direct consequence of certain symmetries of the studied constrained system.…”

“…In this paper we apply a geometric theory of nonholonomic mechanical systems which was developed for the first order in [7] and then was generalized for the higher order case in [8]. The theory enables an investigation of different aspects of these systems; reduced equations of motion of constrained systems (equations of motion on the constraint submanifold), deformed equations of motion (deformation of original unconstrained motion equations by adding Chetaev forces), constraint symmetries and corresponding conservation laws [9] and represents a reasonable concept for an alternative mathematical treatment of concrete examples of such systems, either with linear constraints [5,20] or with nonlinear constraints [10,20,21] and even with the higher order constraints [19].…”