September  2021, 13(3): 477-499. doi: 10.3934/jgm.2021021

Local convexity for second order differential equations on a Lie algebroid

1. 

ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento de Matemáticas, Estadística e IO, Sección de Matemáticas y Física, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain

2. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049 Madrid, Spain

3. 

Departamento de Matemática Aplicada e IUMA, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

Dedicated to the memory of K Mackenzie

Received  March 2021 Published  September 2021 Early access  August 2021

A theory of local convexity for a second order differential equation (${\text{sode}}$) on a Lie algebroid is developed. The particular case when the ${\text{sode}}$ is homogeneous quadratic is extensively discussed.

Citation: Juan Carlos Marrero, David Martín de Diego, Eduardo Martínez. Local convexity for second order differential equations on a Lie algebroid. Journal of Geometric Mechanics, 2021, 13 (3) : 477-499. doi: 10.3934/jgm.2021021
References:
[1]

A. Anahory Simoes, J. C. Marrero and D. Martín de Diego, Exact discrete Lagrangian mechanics for nonholonomic mechanics, preprint, arXiv: 2003.11362, 2020. Google Scholar

[2]

J. CortésM. de LeónJ. C. MarreroD. Martín de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 509-558.  doi: 10.1142/S0219887806001211.  Google Scholar

[3]

J. CortésM. de LeónJ. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Contin. Dyn. Syst., 24 (2009), 213-271.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[4]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[5]

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. doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[6]

M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, Vol. 158. North-Holland Publishing Co., Amsterdam, 1989.  Google Scholar

[7]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520. doi: 10.1063/1.3049752.  Google Scholar

[8]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, Vol. 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[9]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series Vol. 213, Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[10]

J. C. MarreroD. Martín de Diego and E. Martínez, Discrete lagrangian and hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 3003-3004.  doi: 10.1088/0951-7715/19/12/C01.  Google Scholar

[11]

J. C. Marrero, D. Martín de Diego and E. Martínez, The local description of discrete mechanics, Geometry, Mechanics, and Dynamics, Fields Inst. Commun., Springer, New York, 73 (2015), 285–317. doi: 10.1007/978-1-4939-2441-7_13.  Google Scholar

[12]

J. C. Marrero, D. Martín de Diego and E. Martínez, On the exact discrete Lagrangian function for variational integrators: theory and applications, preprint, arXiv: 1608.01586, 2016. Google Scholar

[13]

J. C. Marrero, D. Martín de Diego and E. Martínez, Variational integrators and error analysis for reduced mechanical Lagrangian systems, Work in progress, 2021 Google Scholar

[14]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[15]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320.  doi: 10.1023/A:1011965919259.  Google Scholar

[16]

E. Martínez, Variational calculus on Lie algebroids, ESAIM Control Optim. Calc. Var., 14 (2008), 356-380.  doi: 10.1051/cocv:2007056.  Google Scholar

[17]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.  Google Scholar

[18]

G. W. Patrick and C. Cuell, Error analysis variational integrators of unconstrained Lagrangian systems, Numer. Math., 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3.  Google Scholar

[19]

A. Weinstein, Lagrangian Mechanics and groupoids, Fields Inst. Comm., Amer. Math. Soc., Providence, RI,, 7 (1996), 207-231.  Google Scholar

show all references

References:
[1]

A. Anahory Simoes, J. C. Marrero and D. Martín de Diego, Exact discrete Lagrangian mechanics for nonholonomic mechanics, preprint, arXiv: 2003.11362, 2020. Google Scholar

[2]

J. CortésM. de LeónJ. C. MarreroD. Martín de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 509-558.  doi: 10.1142/S0219887806001211.  Google Scholar

[3]

J. CortésM. de LeónJ. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Contin. Dyn. Syst., 24 (2009), 213-271.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[4]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[5]

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. doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[6]

M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, Vol. 158. North-Holland Publishing Co., Amsterdam, 1989.  Google Scholar

[7]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520. doi: 10.1063/1.3049752.  Google Scholar

[8]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, Vol. 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[9]

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series Vol. 213, Cambridge University Press, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[10]

J. C. MarreroD. Martín de Diego and E. Martínez, Discrete lagrangian and hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 3003-3004.  doi: 10.1088/0951-7715/19/12/C01.  Google Scholar

[11]

J. C. Marrero, D. Martín de Diego and E. Martínez, The local description of discrete mechanics, Geometry, Mechanics, and Dynamics, Fields Inst. Commun., Springer, New York, 73 (2015), 285–317. doi: 10.1007/978-1-4939-2441-7_13.  Google Scholar

[12]

J. C. Marrero, D. Martín de Diego and E. Martínez, On the exact discrete Lagrangian function for variational integrators: theory and applications, preprint, arXiv: 1608.01586, 2016. Google Scholar

[13]

J. C. Marrero, D. Martín de Diego and E. Martínez, Variational integrators and error analysis for reduced mechanical Lagrangian systems, Work in progress, 2021 Google Scholar

[14]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[15]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67 (2001), 295-320.  doi: 10.1023/A:1011965919259.  Google Scholar

[16]

E. Martínez, Variational calculus on Lie algebroids, ESAIM Control Optim. Calc. Var., 14 (2008), 356-380.  doi: 10.1051/cocv:2007056.  Google Scholar

[17]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.  Google Scholar

[18]

G. W. Patrick and C. Cuell, Error analysis variational integrators of unconstrained Lagrangian systems, Numer. Math., 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3.  Google Scholar

[19]

A. Weinstein, Lagrangian Mechanics and groupoids, Fields Inst. Comm., Amer. Math. Soc., Providence, RI,, 7 (1996), 207-231.  Google Scholar

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