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

The Hamilton-Jacobi equation, integrability, and nonholonomic systems

Larry M. Bates 1, , Francesco Fassò 2, and  Nicola Sansonetto 3,

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.
Keywords: complete integrability, nonholonomic mechanics., Hamilton-Jacobi equation.
Mathematics Subject Classification: Primary: 70H20, 37J60, 70F2.
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, Reading, second edition, 1978.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical systems IV, Encyclopaedia Math. Sci. Springer, 4 (2001), 1-138.  Google Scholar

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

[5]

M. Barbero-Liñán, M. de León and D. Martín de Diego, Lagrangian submanifolds and the Hamilton-Jacobi equation, Monatsh. Math., 171 (2013), 269-290. doi: 10.1007/s00605-013-0522-1.  Google Scholar

[6]

L. M. Bates, Examples of singular nonholonomic reduction, Rep. Math. Phys., 42 (1998), 231-247. doi: 10.1016/S0034-4877(98)80012-8.  Google Scholar

[7]

L. M. Bates and R. Cushman, What is a completely integrable nonholonomic dynamical system?, Rep. Math. Phys., 44 (1999), 29-35. doi: 10.1016/S0034-4877(99)80142-6.  Google Scholar

[8]

L. M. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308. doi: 10.1016/0034-4877(96)84069-9.  Google Scholar

[9]

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

[10]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. doi: 10.1007/s002200050412.  Google Scholar

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

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

[13]

R. Cushman, D. Kemppeinen, J. Śniatycki and L. M. Bates, Geometry of nonholonomic constraints, Rep. Math. Phys., 36 (1995), 275-286. doi: 10.1016/0034-4877(96)83625-1.  Google Scholar

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

[15]

L. C. Evans, Weak kam theory and partial differential equations, Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Mathematics, 1927 (2008), 123-154. doi: 10.1007/978-3-540-75914-0_4.  Google Scholar

[16]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[17]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics, Rep. Math. Phys., 62 (2008), 345-367. doi: 10.1016/S0034-4877(09)00005-6.  Google Scholar

[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), 389-416. doi: 10.3934/jgm.2009.1.389.  Google Scholar

[19]

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

[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), 350-356, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533 Kluwer, Dordrecht, 1999.  Google Scholar

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

[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), 072905.  Google Scholar

[23]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Systems and the $N$-body Problem, Applied Mathematical Sciences 90, Springer, second edition, 2009.  Google Scholar

[24]

A. S. Mischenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121. Google Scholar

[25]

N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moskow Math. Soc., 26 (1972), 181-198.  Google Scholar

[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), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[27]

M. Pavon, Hamilton-Jacobi equation for nonholonomic mechanics, J. Math. Phys., 46 (2005), 032902, 8pp. doi: 10.1063/1.1858441.  Google Scholar

[28]

A. van der Schaft and B. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys., 34 (1994), 225-233. doi: 10.1016/0034-4877(94)90038-8.  Google Scholar

[29]

R. van Dooren, Second form of the generalized Hamilton-Jacobi method for nonholonomic dynamical systems, J. Appl. Math. Phys., 29 (1978), 828-834. doi: 10.1007/BF01589294.  Google Scholar

[30]

N. Woodhouse, Geometric Quantization, Oxford mathematical monographs. Oxford university press, second edition, 1991.  Google Scholar

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, second edition, 1978.  Google Scholar

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical systems IV, Encyclopaedia Math. Sci. Springer, 4 (2001), 1-138.  Google Scholar

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

[5]

M. Barbero-Liñán, M. de León and D. Martín de Diego, Lagrangian submanifolds and the Hamilton-Jacobi equation, Monatsh. Math., 171 (2013), 269-290. doi: 10.1007/s00605-013-0522-1.  Google Scholar

[6]

L. M. Bates, Examples of singular nonholonomic reduction, Rep. Math. Phys., 42 (1998), 231-247. doi: 10.1016/S0034-4877(98)80012-8.  Google Scholar

[7]

L. M. Bates and R. Cushman, What is a completely integrable nonholonomic dynamical system?, Rep. Math. Phys., 44 (1999), 29-35. doi: 10.1016/S0034-4877(99)80142-6.  Google Scholar

[8]

L. M. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308. doi: 10.1016/0034-4877(96)84069-9.  Google Scholar

[9]

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

[10]

O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. doi: 10.1007/s002200050412.  Google Scholar

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

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

[13]

R. Cushman, D. Kemppeinen, J. Śniatycki and L. M. Bates, Geometry of nonholonomic constraints, Rep. Math. Phys., 36 (1995), 275-286. doi: 10.1016/0034-4877(96)83625-1.  Google Scholar

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

[15]

L. C. Evans, Weak kam theory and partial differential equations, Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Mathematics, 1927 (2008), 123-154. doi: 10.1007/978-3-540-75914-0_4.  Google Scholar

[16]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[17]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics, Rep. Math. Phys., 62 (2008), 345-367. doi: 10.1016/S0034-4877(09)00005-6.  Google Scholar

[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), 389-416. doi: 10.3934/jgm.2009.1.389.  Google Scholar

[19]

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

[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), 350-356, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533 Kluwer, Dordrecht, 1999.  Google Scholar

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

[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), 072905.  Google Scholar

[23]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Systems and the $N$-body Problem, Applied Mathematical Sciences 90, Springer, second edition, 2009.  Google Scholar

[24]

A. S. Mischenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121. Google Scholar

[25]

N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moskow Math. Soc., 26 (1972), 181-198.  Google Scholar

[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), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[27]

M. Pavon, Hamilton-Jacobi equation for nonholonomic mechanics, J. Math. Phys., 46 (2005), 032902, 8pp. doi: 10.1063/1.1858441.  Google Scholar

[28]

A. van der Schaft and B. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys., 34 (1994), 225-233. doi: 10.1016/0034-4877(94)90038-8.  Google Scholar

[29]

R. van Dooren, Second form of the generalized Hamilton-Jacobi method for nonholonomic dynamical systems, J. Appl. Math. Phys., 29 (1978), 828-834. doi: 10.1007/BF01589294.  Google Scholar

[30]

N. Woodhouse, Geometric Quantization, Oxford mathematical monographs. Oxford university press, second edition, 1991.  Google Scholar

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