# American Institute of Mathematical Sciences

2015, 2015(special): 213-222. doi: 10.3934/proc.2015.0213

## Jacobi fields for second-order differential equations on Lie algebroids

 1 IUMA and Department of Theoretical Physics, University of Zaragoza, Spain 2 Department of Theoretical Physics, University of Zaragoza 3 IUMA and Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza

Received  September 2014 Revised  January 2015 Published  November 2015

We generalize the concept of Jacobi field for general second-order differential equations on a manifold and on a Lie algebroid. The Jacobi equation is expressed in terms of the dynamical covariant derivative and the generalized Jacobi endomorphism associated to the given differential equation.
Citation: José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213
##### References:
 [1] J. F. Cariñena and E. Martínez, Generalized Jacobi equation and Inverse Problem in Classical Mechanics,, In Group Theoretical Methods in Physics II (Moscow 1990), (1990).   Google Scholar [2] J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids,, Discrete and Continuous Dynamical Systems A, 24 2 (2009), 213.  doi: 10.3934/dcds.2009.24.213.  Google Scholar [3] M. Crampin and F. A. E. Pirani, Applicable Differential Geometry,, Cambridge University Press, (1986).   Google Scholar [4] M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005).   Google Scholar [5] P. Foulon, Géométrie des équations différentielles du second ordre,, Ann. Inst. H. Poincaré Phys. Théor., 45 (1986), 1.   Google Scholar [6] L. A. Ibort and E. Martínez, Morse theory for Lagrangian systems,, Unpublished (1997), (1997).   Google Scholar [7] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, Cambridge University Press, (2005).   Google Scholar [8] E. Martínez, Lagrangian Mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.   Google Scholar [9] E. Martínez, Reduction in optimal control theory,, Reports on Mathematical Physics 53 (2004), 53 (2004), 79.  doi: 10.1016/S0034-4877(04)90005-5.  Google Scholar [10] E. Martínez, Classical field theory on Lie algebroids: variational aspects,, J. Phys. A: Math. Gen., 38 (2005), 7145.  doi: 10.1088/0305-4470/38/32/005.  Google Scholar [11] E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 02 (2007), 356.  doi: 10.1051/cocv:2007056.  Google Scholar [12] E. Martínez, J. F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection. Part II,, Differential Geometry and its Applications 3 (1993) 1-29, 3 (1993), 1.   Google Scholar [13] P. W. Michor, The Jacobi flow,, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), 54 (1996), 365.   Google Scholar

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##### References:
 [1] J. F. Cariñena and E. Martínez, Generalized Jacobi equation and Inverse Problem in Classical Mechanics,, In Group Theoretical Methods in Physics II (Moscow 1990), (1990).   Google Scholar [2] J. Cortés, M. de León, J. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids,, Discrete and Continuous Dynamical Systems A, 24 2 (2009), 213.  doi: 10.3934/dcds.2009.24.213.  Google Scholar [3] M. Crampin and F. A. E. Pirani, Applicable Differential Geometry,, Cambridge University Press, (1986).   Google Scholar [4] M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids,, J. Phys. A: Math. Gen., 38 (2005).   Google Scholar [5] P. Foulon, Géométrie des équations différentielles du second ordre,, Ann. Inst. H. Poincaré Phys. Théor., 45 (1986), 1.   Google Scholar [6] L. A. Ibort and E. Martínez, Morse theory for Lagrangian systems,, Unpublished (1997), (1997).   Google Scholar [7] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, Cambridge University Press, (2005).   Google Scholar [8] E. Martínez, Lagrangian Mechanics on Lie algebroids,, Acta Appl. Math., 67 (2001), 295.   Google Scholar [9] E. Martínez, Reduction in optimal control theory,, Reports on Mathematical Physics 53 (2004), 53 (2004), 79.  doi: 10.1016/S0034-4877(04)90005-5.  Google Scholar [10] E. Martínez, Classical field theory on Lie algebroids: variational aspects,, J. Phys. A: Math. Gen., 38 (2005), 7145.  doi: 10.1088/0305-4470/38/32/005.  Google Scholar [11] E. Martínez, Variational calculus on Lie algebroids,, ESAIM: Control, 14 02 (2007), 356.  doi: 10.1051/cocv:2007056.  Google Scholar [12] E. Martínez, J. F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection. Part II,, Differential Geometry and its Applications 3 (1993) 1-29, 3 (1993), 1.   Google Scholar [13] P. W. Michor, The Jacobi flow,, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), 54 (1996), 365.   Google Scholar
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