Jacobi fields in nonholonomic mechanics

In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

[1]  J. Marrero,et al.  The geometry of nonholonomic Chaplygin systems revisited , 2018, Nonlinearity.

[2]  A. D. Lewis,et al.  Variational Principles for Constrained Systems: Theory and Experiment , 1995 .

[3]  Pawel Urbanski,et al.  Algebroids — general differential calculi on vector bundles☆ , 1999 .

[4]  P. Balseiro,et al.  Reduction of nonholonomic systems in two stages and Hamiltonization , 2014, 1409.0456.

[5]  Jerrold E. Marsden,et al.  Geometric mechanics, Lagrangian reduction, and nonholonomic systems , 2001 .

[6]  W. Sarlet,et al.  Derivations of differential forms along the tangent bundle projection , 1992 .

[7]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[8]  Andrew D. Lewis,et al.  Affine connections and distributions with applications to nonholonomic mechanics , 1998 .

[9]  L. Rizzi,et al.  On Jacobi fields and canonical connection in sub-Riemannian geometry , 2015, 1506.01827.

[10]  J. Koiller.  Reduction of some classical non-holonomic systems with symmetry , 1992 .

[11]  Constance de Koning,et al.  Editors , 2003, Annals of Emergency Medicine.

[12]  Derivations of differential forms along the tangent bundle projection , 1992 .

[13]  The Jacobiator of Nonholonomic Systems and the Geometry of Reduced Nonholonomic Brackets , 2013, 1301.1091.

[14]  Jorge Cortés Monforte.  Geometric, control and numerical aspects of nonholonomic systems , 2002 .

[15]  John M. Lee.  Riemannian Manifolds: An Introduction to Curvature , 1997 .

[16]  B. O'neill.  Semi-Riemannian Geometry With Applications to Relativity , 1983 .

[17]  J. Cortes,et al.  On the geometry of generalized Chaplygin systems , 2002, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  Juan Carlos Marrero,et al.  Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic Mechanics , 2008, 0801.4358.

[19]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[20]  Michal Józwikowski.  JACOBI VECTOR FIELDS FOR LAGRANGIAN SYSTEMS ON ALGEBROIDS , 2012, 1205.5952.

[21]  Janusz Grabowski,et al.  Variational calculus with constraints on general algebroids , 2008 .

[22]  Paula Balseiro.  Hamiltonization of Solids of Revolution Through Reduction , 2017, J. Nonlinear Sci..

[23]  A. Mondino.  ON RIEMANNIAN MANIFOLDS , 1999 .

[24]  Kinematic and Geometric Constraints, Servomechanisms and Control of Mechanical Systems , 1996 .

[25]  P. Krishnaprasad,et al.  Nonholonomic mechanical systems with symmetry , 1996 .

[26]  A. Agrachev,et al.  Curvature: A Variational Approach , 2013, Memoirs of the American Mathematical Society.

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

[28]  Alexandre Anahory Simoes,et al.  Exact discrete Lagrangian mechanics for nonholonomic mechanics , 2020, ArXiv.

[29]  J. Grabowski,et al.  Nonholonomic Constraints: a New Viewpoint , 2008, 0806.1117.

[30]  Jean-Marie Souriau,et al.  On Geometric Mechanics , 2007 .

[31]  David Martín de Diego,et al.  Reduction of nonholonomic mechanical systems with symmetries , 1998 .

[32]  R. Bains,et al.  Methods of differential geometry in analytical mechanics , 1992 .

[33]  Sonia Martínez,et al.  Geometric Description of Vakonomic and Nonholonomic Dynamics. Comparison of Solutions , 2002, SIAM J. Control. Optim..

[34]  David Martín de Diego,et al.  On the geometry of non‐holonomic Lagrangian systems , 1996 .

[35]  Geometric , 2021, Encyclopedic Dictionary of Archaeology.

[36]  Juan Carlos Marrero,et al.  Radial kinetic nonholonomic trajectories are Riemannian geodesics! , 2020, Analysis and Mathematical Physics.

[37]  J. L. Synge,et al.  Geodesics in non-holonomic geometry , 1928 .

[38]  J. Cortes,et al.  Nonholonomic Lagrangian systems on Lie algebroids , 2005, math-ph/0512003.

[39]  José F. Cariñena,et al.  Lagrangian systems with constraints: a geometric approach to the method of Lagrange multipliers , 1993 .

[40]  I. Neĭmark,et al.  Dynamics of Nonholonomic Systems , 1972 .

[41]  Juan Carlos Marrero,et al.  Kinematic reduction and the Hamilton-Jacobi equation , 2011, 1110.6066.