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Bundle-theoretic methods for higher-order variational calculus
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 |
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics,, 2nd ed., (1978).
|
[2] |
V. I. Arnold, Mathematical Methods of Classical Mechanics,, Second edition. Graduate Texts in Mathematics, (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.
doi: 10.1088/0951-7715/23/8/006. |
[4] |
L. Bates and J. Sniatycki, Nonholonomic reduction,, Rep. Math. Phys., 32 (1993), 99.
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.
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.
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.
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.
doi: 10.1142/S0219887810004385. |
[9] |
C. Godbillon, Géométrie Différentielle et Mécanique Analytique,, Hermann, (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).
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).
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, (2009), 129.
|
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics,, 2nd ed., (1978).
|
[2] |
V. I. Arnold, Mathematical Methods of Classical Mechanics,, Second edition. Graduate Texts in Mathematics, (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.
doi: 10.1088/0951-7715/23/8/006. |
[4] |
L. Bates and J. Sniatycki, Nonholonomic reduction,, Rep. Math. Phys., 32 (1993), 99.
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.
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.
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.
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.
doi: 10.1142/S0219887810004385. |
[9] |
C. Godbillon, Géométrie Différentielle et Mécanique Analytique,, Hermann, (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).
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).
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, (2009), 129.
|
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
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