December  2014, 6(4): 441-449. doi: 10.3934/jgm.2014.6.441

The Hamilton-Jacobi equation, integrability, and nonholonomic systems

1. 

Department of Mathematics, University of Calgary, Calgary, AB, T2N 1N4, Canada

2. 

Università di Padova, Dipartimento di Matematica Pura e Applicata, Via Trieste 63, 35121 Padova

3. 

Università di Padova, Dipartimento di Matematica, Via Trieste, 63, 35121 Padova, Italy

Received  February 2014 Revised  June 2014 Published  December 2014

By examining the linkage between conservation laws and symmetry, we explain why it appears there should not be an analogue of a complete integral for the Hamilton-Jacobi equation for integrable nonholonomic systems.
Citation: Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics,, Benjamin/Cummings, (1978).

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics,, Graduate Texts in Mathematics 60, (1989). doi: 10.1007/978-1-4757-2063-1.

[3]

V. I. Arnold and A. B. Givental, Symplectic Geometry,, Dynamical systems IV, 4 (2001), 1.

[4]

P. Balseiro, J. C. Marrero, D. Martín de Diego and E. Padrón, A unified framework for mechanics. Hamilton-Jacobi theory and applications,, Nonlinearity, 23 (2010), 1887. doi: 10.1088/0951-7715/23/8/006.

[5]

M. Barbero-Liñán, M. de León and D. Martín de Diego, Lagrangian submanifolds and the Hamilton-Jacobi equation,, Monatsh. Math., 171 (2013), 269. doi: 10.1007/s00605-013-0522-1.

[6]

L. M. Bates, Examples of singular nonholonomic reduction,, Rep. Math. Phys., 42 (1998), 231. doi: 10.1016/S0034-4877(98)80012-8.

[7]

L. M. Bates and R. Cushman, What is a completely integrable nonholonomic dynamical system?,, Rep. Math. Phys., 44 (1999), 29. doi: 10.1016/S0034-4877(99)80142-6.

[8]

L. M. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems,, Rep. Math. Phys., 37 (1996), 295. doi: 10.1016/0034-4877(96)84069-9.

[9]

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

[10]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19. doi: 10.1007/s002200050412.

[11]

J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz Lecanda and N. Román Roy, Geometric Hamilton-Jacobi theory,, Int. J. Geom. Methods Mod. Phys., 3 (2006), 1417. doi: 10.1142/S0219887806001764.

[12]

J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz Lecanda and N. Román Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems,, Int. J. Geom. Methods Mod. Phys., 7 (2010), 431. doi: 10.1142/S0219887810004385.

[13]

R. Cushman, D. Kemppeinen, J. Śniatycki and L. M. Bates, Geometry of nonholonomic constraints,, Rep. Math. Phys., 36 (1995), 275. doi: 10.1016/0034-4877(96)83625-1.

[14]

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.

[15]

L. C. Evans, Weak kam theory and partial differential equations,, Calculus of Variations and Nonlinear Partial Differential Equations, 1927 (2008), 123. doi: 10.1007/978-3-540-75914-0_4.

[16]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93. doi: 10.1007/s10440-005-1139-8.

[17]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics,, Rep. Math. Phys., 62 (2008), 345. doi: 10.1016/S0034-4877(09)00005-6.

[18]

F. Fassò, A. Giacobbe and N. Sansonetto, On the number of weakly Noetherian constants of motion of nonholonomic systems,, J. Geom. Mech., 1 (2009), 389. doi: 10.3934/jgm.2009.1.389.

[19]

A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics,, Cambridge Studies in Advanced Mathematics 88, (2014).

[20]

Y. N. Fedorov, Systems with an invariant measure on Lie groups,, In Hamiltonian systems with three or more degrees of freedom (S'Agarò, (1995), 350.

[21]

D. Iglesias-Ponte, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems,, J. Phys. A: Math. Theor., 41 (2008). doi: 10.1088/1751-8113/41/1/015205.

[22]

M. Leok, T. Ohsawa and D. Sosa, Hamilton-Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints,, J. Math. Phys., 53 (2012).

[23]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Systems and the $N$-body Problem,, Applied Mathematical Sciences 90, (2009).

[24]

A. S. Mischenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems,, Funct. Anal. Appl., 12 (1978), 113.

[25]

N. N. Nekhoroshev, Action-angle variables and their generalizations,, Trans. Moskow Math. Soc., 26 (1972), 181.

[26]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacoby theory via Chaplygin Hamiltonization,, J. Geom. Phys., 61 (2011), 1263. doi: 10.1016/j.geomphys.2011.02.015.

[27]

M. Pavon, Hamilton-Jacobi equation for nonholonomic mechanics,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1858441.

[28]

A. van der Schaft and B. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems,, Rep. Math. Phys., 34 (1994), 225. doi: 10.1016/0034-4877(94)90038-8.

[29]

R. van Dooren, Second form of the generalized Hamilton-Jacobi method for nonholonomic dynamical systems,, J. Appl. Math. Phys., 29 (1978), 828. doi: 10.1007/BF01589294.

[30]

N. Woodhouse, Geometric Quantization,, Oxford mathematical monographs. Oxford university press, (1991).

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics,, Benjamin/Cummings, (1978).

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics,, Graduate Texts in Mathematics 60, (1989). doi: 10.1007/978-1-4757-2063-1.

[3]

V. I. Arnold and A. B. Givental, Symplectic Geometry,, Dynamical systems IV, 4 (2001), 1.

[4]

P. Balseiro, J. C. Marrero, D. Martín de Diego and E. Padrón, A unified framework for mechanics. Hamilton-Jacobi theory and applications,, Nonlinearity, 23 (2010), 1887. doi: 10.1088/0951-7715/23/8/006.

[5]

M. Barbero-Liñán, M. de León and D. Martín de Diego, Lagrangian submanifolds and the Hamilton-Jacobi equation,, Monatsh. Math., 171 (2013), 269. doi: 10.1007/s00605-013-0522-1.

[6]

L. M. Bates, Examples of singular nonholonomic reduction,, Rep. Math. Phys., 42 (1998), 231. doi: 10.1016/S0034-4877(98)80012-8.

[7]

L. M. Bates and R. Cushman, What is a completely integrable nonholonomic dynamical system?,, Rep. Math. Phys., 44 (1999), 29. doi: 10.1016/S0034-4877(99)80142-6.

[8]

L. M. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems,, Rep. Math. Phys., 37 (1996), 295. doi: 10.1016/0034-4877(96)84069-9.

[9]

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

[10]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems,, Comm. Math. Phys., 196 (1998), 19. doi: 10.1007/s002200050412.

[11]

J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz Lecanda and N. Román Roy, Geometric Hamilton-Jacobi theory,, Int. J. Geom. Methods Mod. Phys., 3 (2006), 1417. doi: 10.1142/S0219887806001764.

[12]

J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz Lecanda and N. Román Roy, Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems,, Int. J. Geom. Methods Mod. Phys., 7 (2010), 431. doi: 10.1142/S0219887810004385.

[13]

R. Cushman, D. Kemppeinen, J. Śniatycki and L. M. Bates, Geometry of nonholonomic constraints,, Rep. Math. Phys., 36 (1995), 275. doi: 10.1016/0034-4877(96)83625-1.

[14]

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.

[15]

L. C. Evans, Weak kam theory and partial differential equations,, Calculus of Variations and Nonlinear Partial Differential Equations, 1927 (2008), 123. doi: 10.1007/978-3-540-75914-0_4.

[16]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations,, Acta Appl. Math., 87 (2005), 93. doi: 10.1007/s10440-005-1139-8.

[17]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics,, Rep. Math. Phys., 62 (2008), 345. doi: 10.1016/S0034-4877(09)00005-6.

[18]

F. Fassò, A. Giacobbe and N. Sansonetto, On the number of weakly Noetherian constants of motion of nonholonomic systems,, J. Geom. Mech., 1 (2009), 389. doi: 10.3934/jgm.2009.1.389.

[19]

A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics,, Cambridge Studies in Advanced Mathematics 88, (2014).

[20]

Y. N. Fedorov, Systems with an invariant measure on Lie groups,, In Hamiltonian systems with three or more degrees of freedom (S'Agarò, (1995), 350.

[21]

D. Iglesias-Ponte, M. de León and D. Martín de Diego, Towards a Hamilton-Jacobi theory for nonholonomic mechanical systems,, J. Phys. A: Math. Theor., 41 (2008). doi: 10.1088/1751-8113/41/1/015205.

[22]

M. Leok, T. Ohsawa and D. Sosa, Hamilton-Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints,, J. Math. Phys., 53 (2012).

[23]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Systems and the $N$-body Problem,, Applied Mathematical Sciences 90, (2009).

[24]

A. S. Mischenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems,, Funct. Anal. Appl., 12 (1978), 113.

[25]

N. N. Nekhoroshev, Action-angle variables and their generalizations,, Trans. Moskow Math. Soc., 26 (1972), 181.

[26]

T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacoby theory via Chaplygin Hamiltonization,, J. Geom. Phys., 61 (2011), 1263. doi: 10.1016/j.geomphys.2011.02.015.

[27]

M. Pavon, Hamilton-Jacobi equation for nonholonomic mechanics,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1858441.

[28]

A. van der Schaft and B. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems,, Rep. Math. Phys., 34 (1994), 225. doi: 10.1016/0034-4877(94)90038-8.

[29]

R. van Dooren, Second form of the generalized Hamilton-Jacobi method for nonholonomic dynamical systems,, J. Appl. Math. Phys., 29 (1978), 828. doi: 10.1007/BF01589294.

[30]

N. Woodhouse, Geometric Quantization,, Oxford mathematical monographs. Oxford university press, (1991).

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