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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

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., Benjamin-Cummings, Reading (Ma), 1978.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.

[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), 1887-1918. doi: 10.1088/0951-7715/23/8/006.

[4]

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

[5]

F. Cantrijn, Vector fields generating invariants for classical dissipative systems, J. Math. Phys., 23 (1982), 1589-1595. doi: 10.1063/1.525569.

[6]

F. Cantrijn, M. de León and D. Martín de Diego, On almost-Poisson structures in nonholonomic mechanics, Nonlinearity, 12 (1999), 721-737. doi: 10.1088/0951-7715/12/3/316.

[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), 1417-1458. doi: 10.1142/S0219887806001764.

[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), 431-454. doi: 10.1142/S0219887810004385.

[9]

C. Godbillon, Géométrie Différentielle et Mécanique Analytique, Hermann, Paris, 1969.

[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), 072905 (29 pages). doi: 10.1063/1.4736733.

[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), 015205, 14 pp. doi: 10.1088/1751-8113/41/1/015205.

[12]

M. de León, J. C. Marrero and D. Martín de Diego, A geometric Hamilton-Jacobi theory for classical field theories, In: Variations, geometry and physics, 129-140, Nova Sci. Publ., New York, (2009).

[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), 159-198. doi: 10.3934/jgm.2010.2.159.

[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), 1491-1507. doi: 10.1142/S0219887810004919.

[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), 032902, 32 pp. doi: 10.1063/1.4796088.

[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), 1250074, 24 pp. doi: 10.1142/S0219887812500740.

[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.

[18]

M. de León and P. R. Rodrigues, Methods of differential geometry in analytical mechanics, North-Holland Mathematics Studies, 158. North-Holland Publishing Co., Amsterdam, 1989.

[19]

P. Libermann and Ch.M- Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[20]

J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 605-622. doi: 10.1142/S0219887806001284.

[21]

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

[22]

H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations, Hazell, Watson and Viney Ltd., Aylesbury, Buckinghamshire, U.K. 1966.

[23]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118. Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.

[24]

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.

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin-Cummings, Reading (Ma), 1978.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.

[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), 1887-1918. doi: 10.1088/0951-7715/23/8/006.

[4]

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

[5]

F. Cantrijn, Vector fields generating invariants for classical dissipative systems, J. Math. Phys., 23 (1982), 1589-1595. doi: 10.1063/1.525569.

[6]

F. Cantrijn, M. de León and D. Martín de Diego, On almost-Poisson structures in nonholonomic mechanics, Nonlinearity, 12 (1999), 721-737. doi: 10.1088/0951-7715/12/3/316.

[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), 1417-1458. doi: 10.1142/S0219887806001764.

[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), 431-454. doi: 10.1142/S0219887810004385.

[9]

C. Godbillon, Géométrie Différentielle et Mécanique Analytique, Hermann, Paris, 1969.

[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), 072905 (29 pages). doi: 10.1063/1.4736733.

[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), 015205, 14 pp. doi: 10.1088/1751-8113/41/1/015205.

[12]

M. de León, J. C. Marrero and D. Martín de Diego, A geometric Hamilton-Jacobi theory for classical field theories, In: Variations, geometry and physics, 129-140, Nova Sci. Publ., New York, (2009).

[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), 159-198. doi: 10.3934/jgm.2010.2.159.

[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), 1491-1507. doi: 10.1142/S0219887810004919.

[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), 032902, 32 pp. doi: 10.1063/1.4796088.

[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), 1250074, 24 pp. doi: 10.1142/S0219887812500740.

[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.

[18]

M. de León and P. R. Rodrigues, Methods of differential geometry in analytical mechanics, North-Holland Mathematics Studies, 158. North-Holland Publishing Co., Amsterdam, 1989.

[19]

P. Libermann and Ch.M- Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[20]

J. C. Marrero and D. Sosa, The Hamilton-Jacobi equation on Lie affgebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 605-622. doi: 10.1142/S0219887806001284.

[21]

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

[22]

H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations, Hazell, Watson and Viney Ltd., Aylesbury, Buckinghamshire, U.K. 1966.

[23]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118. Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.

[24]

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.

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