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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
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[1] Bates, L.; Śniatycki, J., Nonholonomic reduction, Rep. math. phys., 32, 99-115, (1992) · Zbl 0798.58026
[2] A.M. Bloch, Nonholonomic Mechanics and Control (with the collaboration of J. Baillieul, P. Crouch, J. Marsden), Interdisc. Appl. Math. 24, Springer-Verlag, New York, 2003.
[3] Bloch, A.M.; Krishnaprasad, P.S.; Marsden, J.E.; Murray, R.M., Nonholonomic mechanical systems with symmetry, Arch. ration. mech. anal., 21-99, (1996) · Zbl 0886.70014
[4] Cantrijn, F., Symplectic approach to nonconservative mechanics, J. math. phys., 23, 1589-1595, (1982)
[5] Cantrijn, F.; de León, M.; Marrero, J.C.; Martín de Diego, D., Reduction of constrained systems with symmetries, J. math. phys., 40, 795-820, (1999) · Zbl 0974.37050
[6] J. Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Math. 1793, Springer-Verlag, Berlin, 2002
[7] Giachetta, G., First integrals of non-holonomic systems and their generators, J. phys. A: math. gen., 33, 5369-5389, (2000) · Zbl 0974.70012
[8] Koiller, J., Reduction of some classical nonholonomic systems with symmetry, Arch. ration. mech. anal., 118, 113-148, (1992) · Zbl 0753.70009
[9] de León, M.; Martín de Diego, D., Solving non-holonomic Lagrangian dynamics in terms of almost product structures, Extracta math., 11, 1-23, (1996)
[10] Martínez, E.; Cari nena, J.F.; Sarlet, W., Derivations of differential forms along the tangent bundle projection, Diff. geometry appl., 2, 17-43, (1992) · Zbl 0748.58002
[11] Martínez, E.; Cari nena, J.F.; Sarlet, W., Derivations of differential forms along the tangent bundle projection II, Diff. geometry appl., 3, 1-29, (1993) · Zbl 0770.53018
[12] Sarlet, W.; Cantrijn, F.; Crampin, M., Pseudo-symmetries, noether’s theorem and the adjoint equation, J. phys. A: math. gen., 20, 1365-1376, (1987) · Zbl 0623.58045
[13] Sarlet, W.; Cantrijn, F.; Saunders, D.J., A geometrical framework for the study of non-holonomic Lagrangian systems, J. phys. A: math. gen., 28, 3253-3268, (1995) · Zbl 0858.70013
[14] Sarlet, W.; Cantrijn, F.; Saunders, D.J., A differential geometric setting for mixed first- and second-order ordinary differential equations, J. phys. A: math. gen., 30, 4031-4052, (1997) · Zbl 0932.37040
[15] Sarlet, W.; Prince, G.E.; Crampin, M., Adjoint symmetries for time-dependent second-order equations, J. phys. A: math. gen., 23, 1335-1347, (1990) · Zbl 0713.58018
[16] Saunders, D.J.; Sarlet, W.; Cantrijn, F., A geometrical framework for the study of non-holonomic Lagrangian systems: II, J. phys. A: math. gen., 29, 4265-4274, (1996) · Zbl 0900.70196
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