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, (1978).   Google Scholar

[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.  doi: 10.1088/0951-7715/23/8/006.  Google Scholar

[3]

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

[4]

A. M. Bloch, "Nonholonomic Mechanics and Control,", Interdisciplinary Applied Mathematics Series \textbf{24}, 24 (2003).  doi: 10.1007/b97376.  Google Scholar

[5]

A. M. Bloch and P. E. Crouch, Representations of Dirac structures on vector spaces and non-linear L-C circuits,, in, 64 (1999), 103.   Google Scholar

[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.  doi: 10.1007/BF02199365.  Google Scholar

[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.  doi: 10.1142/S0219887806001764.  Google Scholar

[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.  doi: 10.1142/S0219887810004385.  Google Scholar

[9]

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

[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.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[11]

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

[12]

T. J. Courant and A. D. Weinstein, "Beyond Poisson Structures,", Travaux en Cours, 27 (1988), 39.   Google Scholar

[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.  doi: 10.1137/S0363012996312039.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

M. Henneaux and C. Teitelboim, "Quantization of Gauge Systems,", Princeton University Press, (1992).   Google Scholar

[19]

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

[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).   Google Scholar

[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.  doi: 10.1080/14689360802294220.  Google Scholar

[22]

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

[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.  doi: 10.1016/S0034-4877(97)85617-0.  Google Scholar

[24]

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

[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).   Google Scholar

[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.  doi: 10.3934/jgm.2010.2.159.  Google Scholar

[27]

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

[28]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005).   Google Scholar

[29]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", second ed., 17 (1999).   Google Scholar

[30]

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

[31]

E. Martínez, Geometric formulation of Mechanics on Lie algebroids,, in, 2 (2001), 209.   Google Scholar

[32]

E. Martínez, Classical field theory on Lie algebroids: Multisymplectic formalism,, preprint \arXiv{math.DG/0411352}., ().   Google Scholar

[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.  doi: 10.1109/81.372847.  Google Scholar

[34]

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

[35]

J. Neimark and N. Fufaev, "Dynamics of Nonholonomic Systems,", Translation of Mathematical Monographs, 33 (1972).   Google Scholar

[36]

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

[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.  doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[38]

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

[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.   Google Scholar

[40]

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

[41]

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

[42]

A. D. Weinstein, Lagrangian Mechanics and groupoids,, in, 7 (1996), 207.   Google Scholar

[43]

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

[44]

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

[45]

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

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", Addison-Wesley, (1978).   Google Scholar

[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.  doi: 10.1088/0951-7715/23/8/006.  Google Scholar

[3]

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

[4]

A. M. Bloch, "Nonholonomic Mechanics and Control,", Interdisciplinary Applied Mathematics Series \textbf{24}, 24 (2003).  doi: 10.1007/b97376.  Google Scholar

[5]

A. M. Bloch and P. E. Crouch, Representations of Dirac structures on vector spaces and non-linear L-C circuits,, in, 64 (1999), 103.   Google Scholar

[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.  doi: 10.1007/BF02199365.  Google Scholar

[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.  doi: 10.1142/S0219887806001764.  Google Scholar

[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.  doi: 10.1142/S0219887810004385.  Google Scholar

[9]

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

[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.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[11]

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

[12]

T. J. Courant and A. D. Weinstein, "Beyond Poisson Structures,", Travaux en Cours, 27 (1988), 39.   Google Scholar

[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.  doi: 10.1137/S0363012996312039.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

M. Henneaux and C. Teitelboim, "Quantization of Gauge Systems,", Princeton University Press, (1992).   Google Scholar

[19]

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

[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).   Google Scholar

[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.  doi: 10.1080/14689360802294220.  Google Scholar

[22]

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

[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.  doi: 10.1016/S0034-4877(97)85617-0.  Google Scholar

[24]

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

[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).   Google Scholar

[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.  doi: 10.3934/jgm.2010.2.159.  Google Scholar

[27]

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

[28]

K. C. H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids,", London Mathematical Society Lecture Note Series, 213 (2005).   Google Scholar

[29]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", second ed., 17 (1999).   Google Scholar

[30]

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

[31]

E. Martínez, Geometric formulation of Mechanics on Lie algebroids,, in, 2 (2001), 209.   Google Scholar

[32]

E. Martínez, Classical field theory on Lie algebroids: Multisymplectic formalism,, preprint \arXiv{math.DG/0411352}., ().   Google Scholar

[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.  doi: 10.1109/81.372847.  Google Scholar

[34]

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

[35]

J. Neimark and N. Fufaev, "Dynamics of Nonholonomic Systems,", Translation of Mathematical Monographs, 33 (1972).   Google Scholar

[36]

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

[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.  doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[38]

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

[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.   Google Scholar

[40]

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

[41]

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

[42]

A. D. Weinstein, Lagrangian Mechanics and groupoids,, in, 7 (1996), 207.   Google Scholar

[43]

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

[44]

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

[45]

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

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