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Adjoint symmetries and the generation of first integrals in non-holonomic mechanics. (English) Zbl 1093.37026
Summary: We discuss a general mechanism by which first integrals of mechanical systems, in particular, systems that satisfy nonholonomic constraints, can be obtained from a systematic search for adjoint symmetries. Such an approach has already been used in our earlier work and is re-advocated here in the context of a recent analysis by G. Giachetta [J. Phys. A, Math. Gen. 33, No. 30, 5369–5389 (2000; Zbl 0974.70012)], in which first integrals are generated by vector fields which are not symmetries. Further advantages of our approach are: the fact that an essential projection operator associated to the constraints need not be related to some given fibre metric on the full evolution space, and the specific selection of a connection, which is naturally associated to this projection and the second-order dynamics on the constraint submanifold. The computational aspects of the method are illustrated by some simple examples.

##### MSC:
 37J60 Nonholonomic dynamical systems 70F25 Nonholonomic systems related to the dynamics of a system of particles 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
##### Keywords:
nonholonomic systems; first integrals; adjoint symmetries
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##### References:
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