December  2012, 4(4): 421-442. doi: 10.3934/jgm.2012.4.421

Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids

1. 

Department of Mathematics, University of California, San Diego, 9500 Gilman Drive La Jolla, CA 92093-0112

2. 

Departamento de Economía Aplicada y Unidad Asociada ULL-CSIC, "Geometría Diferencial y Mecánica Geométrica," Facultad de CC. EE. y Empresariales, Universidad de La Laguna, and Universidad Europea de Canarias, Calle de Inocencio García 1, La Orotava, Tenerife, Canary Islands

Received  June 2012 Revised  November 2012 Published  January 2013

This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\mathcal{T}^E E^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.
Citation: Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421
References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," Addison-Wesley, 2nd edition, 1978.

[2]

P. Balseiro, J. C. Marrero, D. Martín de Diego and E. Padrón, A unified framework for mechanics. Hamilton-Jacobi equation and applications, Nonlinearity, 23 (2010), 1887-1918. doi: 10.1088/0951-7715/23/8/006.

[3]

L. Bates and J. Śniatycki, Nonholonomic reduction, Rep. Math. Phys., 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N.

[4]

A. M. Bloch, "Nonholonomic Mechanics and Control," Interdisciplinary Applied Mathematics Series 24, Springer-Verlag New-York, 2003. doi: 10.1007/b97376.

[5]

A. M. Bloch and P. E. Crouch, Representations of Dirac structures on vector spaces and non-linear L-C circuits, in "Differential Geometry and Control (Boulder, CO, 1997)", Proc. Sympos. Pure Math., 64, AMS, Providence, RI (1999), 103-117.

[6]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.

[7]

J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Munőz Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory, Int. J. Geom. Methods Mod. Phys., 3 (2006), 1417-1458. doi: 10.1142/S0219887806001764.

[8]

J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. C. Munőz Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Methods Mod. Phys., 7 (2010), 431-454. doi: 10.1142/S0219887810004385.

[9]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, IMA J. Math. Control Inform., 21 (2004), 457-492. doi: 10.1093/imamci/21.4.457.

[10]

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

[11]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.2307/2001258.

[12]

T. J. Courant and A. D. Weinstein, "Beyond Poisson Structures," Travaux en Cours, 27, Hermann Paris (1988), 39-49.

[13]

M. Dalsmo and A. J. van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM J. Control Optim., 37 (1998), 54-91. doi: 10.1137/S0363012996312039.

[14]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241. doi: 10.1088/0951-7715/18/5/017.

[15]

M. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399. doi: 10.1063/1.523597.

[16]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253. doi: 10.1016/j.geomphys.2011.06.018.

[17]

K. Grabowska, P. Urbański and J. Grabowski, Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[18]

M. Henneaux and C. Teitelboim, "Quantization of Gauge Systems," Princeton University Press, Princeton, N.J., 1992.

[19]

P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. of Algebra, 129 (1990), 194-230. doi: 10.1016/0021-8693(90)90246-K.

[20]

D. Iglesias-Ponte, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, J. Phys. A, 41 (2008), 015205.

[21]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dyn. Syst., 23 (2008), 351-397. doi: 10.1080/14689360802294220.

[22]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.

[23]

W. S. Koon and J. E. Marsden, The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems, Rep. Math. Phys., 40 (1997), 21-62. doi: 10.1016/S0034-4877(97)85617-0.

[24]

W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134. doi: 10.1016/S0034-4877(98)80007-4.

[25]

M. Leok, T. Ohsawa and D. Sosa, Hamilton-Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints, J. Math. Phys., 53 (2012), 072905 (29 pages).

[26]

M. de León, J. C. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics, J. Geom. Mech., 2 (2010), 159-198. doi: 10.3934/jgm.2010.2.159.

[27]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01.

[28]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series, 213. Cambridge University Press, Cambridge, 2005.

[29]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," second ed., Texts in Applied Mathematics, 17, Springer-Verlag, 1999.

[30]

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

[31]

E. Martínez, Geometric formulation of Mechanics on Lie algebroids, in "Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999", Publ. R. Soc. Mat. Esp., 2 (2001), 209-222.

[32]

E. Martínez, Classical field theory on Lie algebroids: Multisymplectic formalism, preprint arXiv:math.DG/0411352.

[33]

B. M. Maschke, A. J. van der Schaft and P. C. Breedveld, An intrinsic Hamiltonian formulation of the dynamics of LC-circuits, IEEE Trans. Circuits Syst., 42 (1995), 73-82. doi: 10.1109/81.372847.

[34]

T. Mestdag and B. Langerock, A Lie algebroid framework for non-holonomic systems, J. Phys. A, 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011.

[35]

J. Neimark and N. Fufaev, "Dynamics of Nonholonomic Systems," Translation of Mathematical Monographs, 33, AMS, Providence, RI, 1972.

[36]

T. Ohsawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equation and integrability, J. Geom. Mech., 1 (2009), 461-481. doi: 10.3934/jgm.2009.1.461.

[37]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geom. Phys., 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.

[38]

A. J. van der Schaft and B. M. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys., 34 (1994), 225-233. doi: 10.1016/0034-4877(94)90038-8.

[39]

A. J. van der Schaft and B. M. Maschke, The Hamiltonian formulation of energy conserving physical systems with external ports, Arch. Elektr. Übertrag., 49 (1995), 362-371.

[40]

A. M. Vershik and L. D. Fadeev, Lagrangian mechanics in invariant form, Sel. Math. Sov., 1 (1981), 339-350.

[41]

R. W. Weber, Hamiltonian systems with constraints and their meaning in mechanics, Arch. Ration. Mech. Anal., 91 (1986), 309-335. doi: 10.1007/BF00282337.

[42]

A. D. Weinstein, Lagrangian Mechanics and groupoids, in "Mechanics day (Waterloo, ON, 1992)", Fields Inst. Comm., 7 (1996), 207-231.

[43]

H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems, J. Geom. Phys., 57 (1) (2006), 133-156. doi: 10.1016/j.geomphys.2006.02.009.

[44]

H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics. Part II: Variational structures, J. Geom. Phys., 57 (2006), 209-250. doi: 10.1016/j.geomphys.2006.02.012.

[45]

H. Yoshimura and J. E. Marsden, Dirac cotangent bundle reduction, J. Geom. Mech., 1 (2009), 87-158. doi: 10.3934/jgm.2009.1.87.

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," Addison-Wesley, 2nd edition, 1978.

[2]

P. Balseiro, J. C. Marrero, D. Martín de Diego and E. Padrón, A unified framework for mechanics. Hamilton-Jacobi equation and applications, Nonlinearity, 23 (2010), 1887-1918. doi: 10.1088/0951-7715/23/8/006.

[3]

L. Bates and J. Śniatycki, Nonholonomic reduction, Rep. Math. Phys., 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N.

[4]

A. M. Bloch, "Nonholonomic Mechanics and Control," Interdisciplinary Applied Mathematics Series 24, Springer-Verlag New-York, 2003. doi: 10.1007/b97376.

[5]

A. M. Bloch and P. E. Crouch, Representations of Dirac structures on vector spaces and non-linear L-C circuits, in "Differential Geometry and Control (Boulder, CO, 1997)", Proc. Sympos. Pure Math., 64, AMS, Providence, RI (1999), 103-117.

[6]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.

[7]

J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Munőz Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory, Int. J. Geom. Methods Mod. Phys., 3 (2006), 1417-1458. doi: 10.1142/S0219887806001764.

[8]

J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. C. Munőz Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Methods Mod. Phys., 7 (2010), 431-454. doi: 10.1142/S0219887810004385.

[9]

J. Cortés and E. Martínez, Mechanical control systems on Lie algebroids, IMA J. Math. Control Inform., 21 (2004), 457-492. doi: 10.1093/imamci/21.4.457.

[10]

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

[11]

T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.2307/2001258.

[12]

T. J. Courant and A. D. Weinstein, "Beyond Poisson Structures," Travaux en Cours, 27, Hermann Paris (1988), 39-49.

[13]

M. Dalsmo and A. J. van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM J. Control Optim., 37 (1998), 54-91. doi: 10.1137/S0363012996312039.

[14]

Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241. doi: 10.1088/0951-7715/18/5/017.

[15]

M. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399. doi: 10.1063/1.523597.

[16]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253. doi: 10.1016/j.geomphys.2011.06.018.

[17]

K. Grabowska, P. Urbański and J. Grabowski, Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[18]

M. Henneaux and C. Teitelboim, "Quantization of Gauge Systems," Princeton University Press, Princeton, N.J., 1992.

[19]

P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. of Algebra, 129 (1990), 194-230. doi: 10.1016/0021-8693(90)90246-K.

[20]

D. Iglesias-Ponte, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems, J. Phys. A, 41 (2008), 015205.

[21]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dyn. Syst., 23 (2008), 351-397. doi: 10.1080/14689360802294220.

[22]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.

[23]

W. S. Koon and J. E. Marsden, The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems, Rep. Math. Phys., 40 (1997), 21-62. doi: 10.1016/S0034-4877(97)85617-0.

[24]

W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134. doi: 10.1016/S0034-4877(98)80007-4.

[25]

M. Leok, T. Ohsawa and D. Sosa, Hamilton-Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints, J. Math. Phys., 53 (2012), 072905 (29 pages).

[26]

M. de León, J. C. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics, J. Geom. Mech., 2 (2010), 159-198. doi: 10.3934/jgm.2010.2.159.

[27]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308. doi: 10.1088/0305-4470/38/24/R01.

[28]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids," London Mathematical Society Lecture Note Series, 213. Cambridge University Press, Cambridge, 2005.

[29]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," second ed., Texts in Applied Mathematics, 17, Springer-Verlag, 1999.

[30]

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

[31]

E. Martínez, Geometric formulation of Mechanics on Lie algebroids, in "Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999", Publ. R. Soc. Mat. Esp., 2 (2001), 209-222.

[32]

E. Martínez, Classical field theory on Lie algebroids: Multisymplectic formalism, preprint arXiv:math.DG/0411352.

[33]

B. M. Maschke, A. J. van der Schaft and P. C. Breedveld, An intrinsic Hamiltonian formulation of the dynamics of LC-circuits, IEEE Trans. Circuits Syst., 42 (1995), 73-82. doi: 10.1109/81.372847.

[34]

T. Mestdag and B. Langerock, A Lie algebroid framework for non-holonomic systems, J. Phys. A, 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011.

[35]

J. Neimark and N. Fufaev, "Dynamics of Nonholonomic Systems," Translation of Mathematical Monographs, 33, AMS, Providence, RI, 1972.

[36]

T. Ohsawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equation and integrability, J. Geom. Mech., 1 (2009), 461-481. doi: 10.3934/jgm.2009.1.461.

[37]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geom. Phys., 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.

[38]

A. J. van der Schaft and B. M. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys., 34 (1994), 225-233. doi: 10.1016/0034-4877(94)90038-8.

[39]

A. J. van der Schaft and B. M. Maschke, The Hamiltonian formulation of energy conserving physical systems with external ports, Arch. Elektr. Übertrag., 49 (1995), 362-371.

[40]

A. M. Vershik and L. D. Fadeev, Lagrangian mechanics in invariant form, Sel. Math. Sov., 1 (1981), 339-350.

[41]

R. W. Weber, Hamiltonian systems with constraints and their meaning in mechanics, Arch. Ration. Mech. Anal., 91 (1986), 309-335. doi: 10.1007/BF00282337.

[42]

A. D. Weinstein, Lagrangian Mechanics and groupoids, in "Mechanics day (Waterloo, ON, 1992)", Fields Inst. Comm., 7 (1996), 207-231.

[43]

H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems, J. Geom. Phys., 57 (1) (2006), 133-156. doi: 10.1016/j.geomphys.2006.02.009.

[44]

H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics. Part II: Variational structures, J. Geom. Phys., 57 (2006), 209-250. doi: 10.1016/j.geomphys.2006.02.012.

[45]

H. Yoshimura and J. E. Marsden, Dirac cotangent bundle reduction, J. Geom. Mech., 1 (2009), 87-158. doi: 10.3934/jgm.2009.1.87.

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