Adjoint symmetries and the generation of first integrals in non-holonomic mechanics.

*(English)*Zbl 1093.37026Summary: 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 |

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\textit{W. Sarlet} et al., J. Geom. Phys. 55, No. 2, 207--225 (2005; Zbl 1093.37026)

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