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June  2012, 4(2): 129-136. doi: 10.3934/jgm.2012.4.129

Linear weakly Noetherian constants of motion are horizontal gauge momenta

Francesco Fassò 1, , Andrea Giacobbe 1, and  Nicola Sansonetto 2,

1. 

Università di Padova, Dipartimento di Matematica Pura e Applicata, Via Trieste 63, 35121 Padova
2. 

Università di Verona, Dipartimento di Informatica, Cà Vignal 2, Strada Le Grazie 15, 37134 Verona

Received  October 2010 Revised  March 2011 Published  August 2012

noindent The notion of gauge momenta is a generalization of the momentum map which is relevant for nonholonomic systems with symmetry. Weakly Noetherian functions are functions which are constants of motion of all 'natural' nonholonomic systems with a given kinetic energy and any $G$-invariant potential energy. We show that, when the action of the symmetry group on the configuration manifold is free and proper, a function which is linear in the velocities is weakly-Noetherian if anf only if it is a gauge momenta which has a horizontal generator.
Keywords: Noether theorem., constants of motion, first integrals, Nonholonomic systems.
Mathematics Subject Classification: 37J15, 37J05, 70F2.
Citation: Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. Linear weakly Noetherian constants of motion are horizontal gauge momenta. Journal of Geometric Mechanics, 2012, 4 (2) : 129-136. doi: 10.3934/jgm.2012.4.129
References:
[1]

C. Agostinelli, Nuova forma sintetica delle equazioni del moto di un sistema anolonomo ed esistenza di un integrale lineare nelle velocità lagrangiane, Boll. Un. Mat. Ital. (3), 11 (1956), 1-9.

[2]

L. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308. doi: 10.1016/0034-4877(96)84069-9.

[3]

S. Benenti, A 'user-friendly' approach to the dynamical equations of non-holonomic systems, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 036, 33 pp.

[4]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.

[5]

A. M. Bloch, "Nonholonomic Mechanics and Controls,'' Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[6]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in non-holonomic systems, Dynamical Systems, 24 (2009), 187-222. doi: 10.1080/14689360802609344.

[7]

F. Cantrjn, M. de León, M. de Diego and J. Marrero, Reduction of nonholonomic mechanical systems with symmetries, Rep. Math. Phys., 42 (1998), 25-45. doi: 10.1016/S0034-4877(98)80003-7.

[8]

F. Cantrjn, J. Cortés, M. de León and M. de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Phil. Soc., 132 (2002), 323-351.

[9]

R. Cushman, D. Kemppainen, J. Śniatycki and L. Bates, Geometry of nonholonomic constraints, Proceedings of the XXVII Symposium on Mathematical Physics (Toruń, 1994), Rep. Math. Phys., 36 (1995), 275-286. doi: 10.1016/0034-4877(96)83625-1.

[10]

M. de León, J. C. Marrero and D. Martín de Diego, Mechanical systems with nonlinear constraints, Int. J. Th. Phys., 36 (1997), 979-995. doi: 10.1007/BF02435796.

[11]

F. Fassò, A. Ramos and N. Sansonetto, The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions, Reg. Ch. Dyn., 12 (2007), 579-588. doi: 10.1134/S1560354707060019.

[12]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics, Rep. Math. Phys., 62 (2008), 345-367. doi: 10.1016/S0034-4877(09)00005-6.

[13]

F. Fassò, A. Giacobbe and N. Sansonetto, On the number of weakly Noetherian constants of motion of nonholonomic systems, J. Geom. Mech., 1 (2009), 389-416.

[14]

Il. Iliev and Khr. Semerdzhiev, Relations between the first integrals of a nonholonomic mechanical system and of the corresponding system freed of constraints, J. Appl. Math. Mech., 36 (1972), 381-388. doi: 10.1016/0021-8928(72)90049-4.

[15]

Il. Iliev and P. Ilija, On first integrals of a nonholonomic mechanical system, J. Appl. Math. Mech., 39 (1975), 147-150. doi: 10.1016/0021-8928(75)90046-5.

[16]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rat. Mech. An., 118 (1992), 113-148. doi: 10.1007/BF00375092.

[17]

C.-M. Marle, On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints, in "Classical and Quantum Integrability'' (Warsaw, 2001), Banach Center Publ., 59, Polish Acad. Sci., Warsaw, (2003), 223-242

[18]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,'' Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004.

[19]

J. Śniatycki, Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys., 42 (1998), 5-23. doi: 10.1016/S0034-4877(98)80002-5.

[20]

D. V. Zenkov, Linear conservation laws of nonholonomic systems with symmetry, in "Dynamical systems and differential equations'' (Wilmington, NC, 2002), Discrete Contin. Dyn. Syst., 2003, suppl., 967-976.

show all references

References:
[1]

C. Agostinelli, Nuova forma sintetica delle equazioni del moto di un sistema anolonomo ed esistenza di un integrale lineare nelle velocità lagrangiane, Boll. Un. Mat. Ital. (3), 11 (1956), 1-9.

[2]

L. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308. doi: 10.1016/0034-4877(96)84069-9.

[3]

S. Benenti, A 'user-friendly' approach to the dynamical equations of non-holonomic systems, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 036, 33 pp.

[4]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.

[5]

A. M. Bloch, "Nonholonomic Mechanics and Controls,'' Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.

[6]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in non-holonomic systems, Dynamical Systems, 24 (2009), 187-222. doi: 10.1080/14689360802609344.

[7]

F. Cantrjn, M. de León, M. de Diego and J. Marrero, Reduction of nonholonomic mechanical systems with symmetries, Rep. Math. Phys., 42 (1998), 25-45. doi: 10.1016/S0034-4877(98)80003-7.

[8]

F. Cantrjn, J. Cortés, M. de León and M. de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Phil. Soc., 132 (2002), 323-351.

[9]

R. Cushman, D. Kemppainen, J. Śniatycki and L. Bates, Geometry of nonholonomic constraints, Proceedings of the XXVII Symposium on Mathematical Physics (Toruń, 1994), Rep. Math. Phys., 36 (1995), 275-286. doi: 10.1016/0034-4877(96)83625-1.

[10]

M. de León, J. C. Marrero and D. Martín de Diego, Mechanical systems with nonlinear constraints, Int. J. Th. Phys., 36 (1997), 979-995. doi: 10.1007/BF02435796.

[11]

F. Fassò, A. Ramos and N. Sansonetto, The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions, Reg. Ch. Dyn., 12 (2007), 579-588. doi: 10.1134/S1560354707060019.

[12]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics, Rep. Math. Phys., 62 (2008), 345-367. doi: 10.1016/S0034-4877(09)00005-6.

[13]

F. Fassò, A. Giacobbe and N. Sansonetto, On the number of weakly Noetherian constants of motion of nonholonomic systems, J. Geom. Mech., 1 (2009), 389-416.

[14]

Il. Iliev and Khr. Semerdzhiev, Relations between the first integrals of a nonholonomic mechanical system and of the corresponding system freed of constraints, J. Appl. Math. Mech., 36 (1972), 381-388. doi: 10.1016/0021-8928(72)90049-4.

[15]

Il. Iliev and P. Ilija, On first integrals of a nonholonomic mechanical system, J. Appl. Math. Mech., 39 (1975), 147-150. doi: 10.1016/0021-8928(75)90046-5.

[16]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rat. Mech. An., 118 (1992), 113-148. doi: 10.1007/BF00375092.

[17]

C.-M. Marle, On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints, in "Classical and Quantum Integrability'' (Warsaw, 2001), Banach Center Publ., 59, Polish Acad. Sci., Warsaw, (2003), 223-242

[18]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,'' Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004.

[19]

J. Śniatycki, Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys., 42 (1998), 5-23. doi: 10.1016/S0034-4877(98)80002-5.

[20]

D. V. Zenkov, Linear conservation laws of nonholonomic systems with symmetry, in "Dynamical systems and differential equations'' (Wilmington, NC, 2002), Discrete Contin. Dyn. Syst., 2003, suppl., 967-976.

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