A Hamilton-Jacobi theory on Poisson manifolds

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March 2014, 6(1): 121-140. doi: 10.3934/jgm.2014.6.121

A Hamilton-Jacobi theory on Poisson manifolds

Manuel de León ^1, , David Martín de Diego ^1, and Miguel Vaquero ^1,

1.	Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), c\ Nicolás Cabrera, n 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain, Spain, Spain

Received November 2012 Revised January 2014 Published April 2014

Abstract
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In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.

Keywords: time-dependent systems, external forces., Poisson manifolds, Hamilton-Jacobi theory, nonholonomic mechanics.

Mathematics Subject Classification: Primary: 70H20, 70 G45, 70F25, 37J6.

Citation: Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, 2nd ed., (1978). Google Scholar
[2]	V. I. Arnold, Mathematical Methods of Classical Mechanics,, Second edition. Graduate Texts in Mathematics, (1989). Google Scholar
[3]	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
[4]	L. Bates and J. Sniatycki, Nonholonomic reduction,, Rep. Math. Phys., 32 (1993), 99. doi: 10.1016/0034-4877(93)90073-N. Google Scholar
[5]	F. Cantrijn, Vector fields generating invariants for classical dissipative systems,, J. Math. Phys., 23 (1982), 1589. doi: 10.1063/1.525569. Google Scholar
[6]	F. Cantrijn, M. de León and D. Martín de Diego, On almost-Poisson structures in nonholonomic mechanics,, Nonlinearity, 12 (1999), 721. doi: 10.1088/0951-7715/12/3/316. Google Scholar
[7]	J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 1417. doi: 10.1142/S0219887806001764. Google Scholar
[8]	J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems,, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 431. doi: 10.1142/S0219887810004385. Google Scholar
[9]	C. Godbillon, Géométrie Différentielle et Mécanique Analytique,, Hermann, (1969). Google Scholar
[10]	M. Leok, T. Ohsawa and D. Sosa, Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints,, Journal of Mathematical Physics, 53 (2012). doi: 10.1063/1.4736733. Google Scholar
[11]	M. de León, D. Iglesias-Ponte and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems,, Journal of Physics A: Math. Gen., 41 (2008). doi: 10.1088/1751-8113/41/1/015205. Google Scholar
[12]	M. de León, J. C. Marrero and D. Martín de Diego, A geometric Hamilton-Jacobi theory for classical field theories,, In: Variations, (2009), 129. Google Scholar
[13]	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
[14]	M. de León, D. Martín de Diego, J. C. Marrero, M. Salgado and S. Vilariño, Hamilton-Jacobi theory in $k$-symplectic field theories,, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1491. doi: 10.1142/S0219887810004919. Google Scholar
[15]	M. de León, J. C. Marrero, D. Martín de Diego and M. Vaquero, A Hamilton-Jacobi theory for singular lagrangian systems,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4796088. Google Scholar
[16]	M. de León, D. Martín de Diego and M. Vaquero, A Hamilton-Jacobi theory for singular lagrangian systems in the Skinner and Rusk setting,, Int. J. Geom. Meth. Mod. Phys., 9 (2012). doi: 10.1142/S0219887812500740. Google Scholar
[17]	M. de León, D. Martín de Diego, C. Martínez-Campos and M. Vaquero, A Hamilton-Jacobi theory in infinite dimensional phase spaces,, In preparation., (). Google Scholar
[18]	M. de León and P. R. Rodrigues, Methods of differential geometry in analytical mechanics,, North-Holland Mathematics Studies, (1989). Google Scholar
[19]	P. Libermann and Ch.M- Marle, Symplectic Geometry and Analytical Mechanics,, D. Reidel Publishing Co., (1987). doi: 10.1007/978-94-009-3807-6. Google Scholar
[20]	J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids,, Int. J. Geom. Methods Mod. Phys., 3 (2006), 605. doi: 10.1142/S0219887806001284. Google Scholar
[21]	T. Oshawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equations and integrability,, J. Geom. Mech., 1 (2009), 461. doi: 10.3934/jgm.2009.1.461. Google Scholar
[22]	H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations,, Hazell, (1966). Google Scholar
[23]	I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Progress in Mathematics, (1994). doi: 10.1007/978-3-0348-8495-2. Google Scholar
[24]	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

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References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, 2nd ed., (1978). Google Scholar
[2]	V. I. Arnold, Mathematical Methods of Classical Mechanics,, Second edition. Graduate Texts in Mathematics, (1989). Google Scholar
[3]	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
[4]	L. Bates and J. Sniatycki, Nonholonomic reduction,, Rep. Math. Phys., 32 (1993), 99. doi: 10.1016/0034-4877(93)90073-N. Google Scholar
[5]	F. Cantrijn, Vector fields generating invariants for classical dissipative systems,, J. Math. Phys., 23 (1982), 1589. doi: 10.1063/1.525569. Google Scholar
[6]	F. Cantrijn, M. de León and D. Martín de Diego, On almost-Poisson structures in nonholonomic mechanics,, Nonlinearity, 12 (1999), 721. doi: 10.1088/0951-7715/12/3/316. Google Scholar
[7]	J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory,, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 1417. doi: 10.1142/S0219887806001764. Google Scholar
[8]	J. F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems,, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 431. doi: 10.1142/S0219887810004385. Google Scholar
[9]	C. Godbillon, Géométrie Différentielle et Mécanique Analytique,, Hermann, (1969). Google Scholar
[10]	M. Leok, T. Ohsawa and D. Sosa, Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints,, Journal of Mathematical Physics, 53 (2012). doi: 10.1063/1.4736733. Google Scholar
[11]	M. de León, D. Iglesias-Ponte and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems,, Journal of Physics A: Math. Gen., 41 (2008). doi: 10.1088/1751-8113/41/1/015205. Google Scholar
[12]	M. de León, J. C. Marrero and D. Martín de Diego, A geometric Hamilton-Jacobi theory for classical field theories,, In: Variations, (2009), 129. Google Scholar
[13]	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
[14]	M. de León, D. Martín de Diego, J. C. Marrero, M. Salgado and S. Vilariño, Hamilton-Jacobi theory in $k$-symplectic field theories,, Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1491. doi: 10.1142/S0219887810004919. Google Scholar
[15]	M. de León, J. C. Marrero, D. Martín de Diego and M. Vaquero, A Hamilton-Jacobi theory for singular lagrangian systems,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4796088. Google Scholar
[16]	M. de León, D. Martín de Diego and M. Vaquero, A Hamilton-Jacobi theory for singular lagrangian systems in the Skinner and Rusk setting,, Int. J. Geom. Meth. Mod. Phys., 9 (2012). doi: 10.1142/S0219887812500740. Google Scholar
[17]	M. de León, D. Martín de Diego, C. Martínez-Campos and M. Vaquero, A Hamilton-Jacobi theory in infinite dimensional phase spaces,, In preparation., (). Google Scholar
[18]	M. de León and P. R. Rodrigues, Methods of differential geometry in analytical mechanics,, North-Holland Mathematics Studies, (1989). Google Scholar
[19]	P. Libermann and Ch.M- Marle, Symplectic Geometry and Analytical Mechanics,, D. Reidel Publishing Co., (1987). doi: 10.1007/978-94-009-3807-6. Google Scholar
[20]	J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids,, Int. J. Geom. Methods Mod. Phys., 3 (2006), 605. doi: 10.1142/S0219887806001284. Google Scholar
[21]	T. Oshawa and A. M. Bloch, Nonholonomic Hamilton-Jacobi equations and integrability,, J. Geom. Mech., 1 (2009), 461. doi: 10.3934/jgm.2009.1.461. Google Scholar
[22]	H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations,, Hazell, (1966). Google Scholar
[23]	I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Progress in Mathematics, (1994). doi: 10.1007/978-3-0348-8495-2. Google Scholar
[24]	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

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