Poincaré transformations in nonholonomic mechanics

Abstract We report on the application of the Poincare transformation (from the theory of adaptive geometric integrators) to nonholonomic systems—mechanical systems with non-integrable velocity constraints. We prove that this transformation can be used to express the dynamics of certain nonholonomic systems at a fixed energy value in Hamiltonian form; examples and potential applications are also discussed.

[1]  D. Tompsett.  Conservation laws , 1987 .

[2]  O. Fernandez.  Quantizing conditionally variational nonholonomic systems , 2014 .

[3]  M. Humphry,et al.  Bond breaking coupled with translation in rolling of covalently bound molecules. , 2005, Physical review letters.

[4]  Y. Kivshar,et al.  Transport of fullerene molecules along graphene nanoribbons , 2012, Scientific Reports.

[5]  A. Borisov,et al.  Isomorphism and Hamilton representation of some nonholonomic systems , 2007 .

[6]  Jair Koiller,et al.  Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization , 2005 .

[7]  B. Leimkuhler,et al.  Simulating Hamiltonian Dynamics: Hamiltonian PDEs , 2005 .

[8]  Alexey V. Borisov,et al.  Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems , 2008 .

[9]  Marco Favretti,et al.  Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints , 1998 .

[10]  T. Mestdag,et al.  Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations , 2008, 0812.0437.

[11]  S. A. Chaplygin.  On the theory of motion of nonholonomic systems. The reducing-multiplier theorem , 2008 .

[12]  Reduction of nonholonomic systems in two stages , 2014 .

[13]  P. Balseiro,et al.  Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems , 2011, 1104.0880.

[14]  Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere , 2008, 0808.0854.

[15]  T. Mestdag,et al.  A generalization of Chaplygin’s Reducibility Theorem , 2009, 0909.4018.

[16]  A. Bloch,et al.  Nonholonomic Mechanics and Control , 2004, IEEE Transactions on Automatic Control.

[17]  Anthony M. Bloch,et al.  Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data , 2008 .