American Institute of Mathematical Sciences

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

Jacobi fields for second-order differential equations on Lie algebroids

José F. Cariñena 1, , Irina Gheorghiu 2, and  Eduardo Martínez 3,

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.
Keywords: second-order di erential equations, Jacobi fi elds, Jacobi equation., Lie algebroids.
Mathematics Subject Classification: 34A26, 58Z05, 58Cx.
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), Nova Science Publishers, New York, 1991 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), 213-271 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), R241-R308  Google Scholar

[5]

P. Foulon, Géométrie des équations différentielles du second ordre, Ann. Inst. H. Poincaré Phys. Théor., 45 (1986), 1, 1-28  Google Scholar

[6]

L. A. Ibort and E. Martínez, Morse theory for Lagrangian systems, Unpublished (1997) Google Scholar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), 295-320  Google Scholar

[9]

E. Martínez, Reduction in optimal control theory, Reports on Mathematical Physics 53 (2004), 79-90 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), 7145-7160 doi: 10.1088/0305-4470/38/32/005.  Google Scholar

[11]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations 14 02, 2007, 356-380 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  Google Scholar

[13]

P. W. Michor, The Jacobi flow, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), 365-372  Google Scholar

show all references

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), Nova Science Publishers, New York, 1991 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), 213-271 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), R241-R308  Google Scholar

[5]

P. Foulon, Géométrie des équations différentielles du second ordre, Ann. Inst. H. Poincaré Phys. Théor., 45 (1986), 1, 1-28  Google Scholar

[6]

L. A. Ibort and E. Martínez, Morse theory for Lagrangian systems, Unpublished (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), 295-320  Google Scholar

[9]

E. Martínez, Reduction in optimal control theory, Reports on Mathematical Physics 53 (2004), 79-90 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), 7145-7160 doi: 10.1088/0305-4470/38/32/005.  Google Scholar

[11]

E. Martínez, Variational calculus on Lie algebroids, ESAIM: Control, Optimisation and Calculus of Variations 14 02, 2007, 356-380 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  Google Scholar

[13]

P. W. Michor, The Jacobi flow, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), 365-372  Google Scholar

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