American Institute of Mathematical Sciences

Advanced
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]

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

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

[1]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007
[2]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003
[3]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405
[4]

Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021044
[5]

Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021062
[6]

Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015
[7]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[8]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011
[9]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002
[10]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001
[11]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451
[12]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
[13]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039
[14]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2559-2599. doi: 10.3934/dcds.2020375
[15]

John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004
[16]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002
[17]

Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021054
[18]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[19]

Elimhan N. Mahmudov. Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021010
[20]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences