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
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show all references

References:
[1]

Addison-Wesley, 2nd edition, 1978.  Google Scholar

[2]

Nonlinearity, 23 (2010), 1887-1918. doi: 10.1088/0951-7715/23/8/006.  Google Scholar

[3]

Rep. Math. Phys., 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N.  Google Scholar

[4]

Interdisciplinary Applied Mathematics Series 24, Springer-Verlag New-York, 2003. doi: 10.1007/b97376.  Google Scholar

[5]

in "Differential Geometry and Control (Boulder, CO, 1997)", Proc. Sympos. Pure Math., 64, AMS, Providence, RI (1999), 103-117.  Google Scholar

[6]

Arch. Rational Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[7]

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[8]

Int. J. Geom. Methods Mod. Phys., 7 (2010), 431-454. doi: 10.1142/S0219887810004385.  Google Scholar

[9]

IMA J. Math. Control Inform., 21 (2004), 457-492. doi: 10.1093/imamci/21.4.457.  Google Scholar

[10]

Discrete Contin. Dyn. Syst., 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.  Google Scholar

[11]

Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.2307/2001258.  Google Scholar

[12]

Travaux en Cours, 27, Hermann Paris (1988), 39-49.  Google Scholar

[13]

SIAM J. Control Optim., 37 (1998), 54-91. doi: 10.1137/S0363012996312039.  Google Scholar

[14]

Nonlinearity, 18 (2005), 2211-2241. doi: 10.1088/0951-7715/18/5/017.  Google Scholar

[15]

J. Math. Phys., 19 (1978), 2388-2399. doi: 10.1063/1.523597.  Google Scholar

[16]

J. Geom. Phys., 61 (2011), 2233-2253. doi: 10.1016/j.geomphys.2011.06.018.  Google Scholar

[17]

Int. J. Geom. Methods Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.  Google Scholar

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Princeton University Press, Princeton, N.J., 1992.  Google Scholar

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[22]

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[31]

in "Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999", Publ. R. Soc. Mat. Esp., 2 (2001), 209-222.  Google Scholar

[32]

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

[33]

IEEE Trans. Circuits Syst., 42 (1995), 73-82. doi: 10.1109/81.372847.  Google Scholar

[34]

J. Phys. A, 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011.  Google Scholar

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Translation of Mathematical Monographs, 33, AMS, Providence, RI, 1972. Google Scholar

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[37]

J. Geom. Phys., 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[38]

Rep. Math. Phys., 34 (1994), 225-233. doi: 10.1016/0034-4877(94)90038-8.  Google Scholar

[39]

Arch. Elektr. Übertrag., 49 (1995), 362-371. Google Scholar

[40]

Sel. Math. Sov., 1 (1981), 339-350. Google Scholar

[41]

Arch. Ration. Mech. Anal., 91 (1986), 309-335. doi: 10.1007/BF00282337.  Google Scholar

[42]

in "Mechanics day (Waterloo, ON, 1992)", Fields Inst. Comm., 7 (1996), 207-231.  Google Scholar

[43]

J. Geom. Phys., 57 (1) (2006), 133-156. doi: 10.1016/j.geomphys.2006.02.009.  Google Scholar

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J. Geom. Phys., 57 (2006), 209-250. doi: 10.1016/j.geomphys.2006.02.012.  Google Scholar

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J. Geom. Mech., 1 (2009), 87-158. doi: 10.3934/jgm.2009.1.87.  Google Scholar

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