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

A Hamilton-Jacobi theory on Poisson manifolds

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.
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:
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In: Variations, geometry and physics, 129-140, Nova Sci. Publ., New York, (2009).  Google Scholar

[13]

J. Geom. Mech., 2 (2010), 159-198. doi: 10.3934/jgm.2010.2.159.  Google Scholar

[14]

Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1491-1507. doi: 10.1142/S0219887810004919.  Google Scholar

[15]

J. Math. Phys., 54 (2013), 032902, 32 pp. doi: 10.1063/1.4796088.  Google Scholar

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Int. J. Geom. Meth. Mod. Phys., 9 (2012), 1250074, 24 pp. 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]

North-Holland Mathematics Studies, 158. North-Holland Publishing Co., Amsterdam, 1989.  Google Scholar

[19]

D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

Int. J. Geom. Methods Mod. Phys., 3 (2006), 605-622. doi: 10.1142/S0219887806001284.  Google Scholar

[21]

J. Geom. Mech., 1 (2009), 461-481. doi: 10.3934/jgm.2009.1.461.  Google Scholar

[22]

Hazell, Watson and Viney Ltd., Aylesbury, Buckinghamshire, U.K. 1966. Google Scholar

[23]

Progress in Mathematics, 118. Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

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

show all references

References:
[1]

2nd ed., Benjamin-Cummings, Reading (Ma), 1978.  Google Scholar

[2]

Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.  Google Scholar

[3]

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

[4]

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

[5]

J. Math. Phys., 23 (1982), 1589-1595. doi: 10.1063/1.525569.  Google Scholar

[6]

Nonlinearity, 12 (1999), 721-737. doi: 10.1088/0951-7715/12/3/316.  Google Scholar

[7]

Int. J. Geom. Meth. Mod. Phys., 3 (2006), 1417-1458. doi: 10.1142/S0219887806001764.  Google Scholar

[8]

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

[9]

Hermann, Paris, 1969.  Google Scholar

[10]

Journal of Mathematical Physics, 53 (2012), 072905 (29 pages). doi: 10.1063/1.4736733.  Google Scholar

[11]

Journal of Physics A: Math. Gen., 41 (2008), 015205, 14 pp. doi: 10.1088/1751-8113/41/1/015205.  Google Scholar

[12]

In: Variations, geometry and physics, 129-140, Nova Sci. Publ., New York, (2009).  Google Scholar

[13]

J. Geom. Mech., 2 (2010), 159-198. doi: 10.3934/jgm.2010.2.159.  Google Scholar

[14]

Int. J. Geom. Meth. Mod. Phys., 7 (2010), 1491-1507. doi: 10.1142/S0219887810004919.  Google Scholar

[15]

J. Math. Phys., 54 (2013), 032902, 32 pp. doi: 10.1063/1.4796088.  Google Scholar

[16]

Int. J. Geom. Meth. Mod. Phys., 9 (2012), 1250074, 24 pp. 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]

North-Holland Mathematics Studies, 158. North-Holland Publishing Co., Amsterdam, 1989.  Google Scholar

[19]

D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

Int. J. Geom. Methods Mod. Phys., 3 (2006), 605-622. doi: 10.1142/S0219887806001284.  Google Scholar

[21]

J. Geom. Mech., 1 (2009), 461-481. doi: 10.3934/jgm.2009.1.461.  Google Scholar

[22]

Hazell, Watson and Viney Ltd., Aylesbury, Buckinghamshire, U.K. 1966. Google Scholar

[23]

Progress in Mathematics, 118. Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[24]

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

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