The Hamilton-Jacobi equation, integrability, and nonholonomic systems

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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

Abstract
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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.
[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.
[3]	V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical systems IV, Encyclopaedia Math. Sci. Springer, 4 (2001), 1-138.
[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.
[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.
[6]	L. M. Bates, Examples of singular nonholonomic reduction, Rep. Math. Phys., 42 (1998), 231-247. 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), 29-35. 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), 295-308. doi: 10.1016/0034-4877(96)84069-9.
[9]	L. M. Bates and J. Śniatycki, Nonholonomic reduction, Rep. Math. Phys., 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N.
[10]	O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. 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), 1417-1458. 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), 431-454. doi: 10.1142/S0219887810004385.
[13]	R. Cushman, D. Kemppeinen, J. Śniatycki and L. M. Bates, Geometry of nonholonomic constraints, Rep. Math. Phys., 36 (1995), 275-286. 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), 159-198. 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, Lecture Notes in Mathematics, 1927 (2008), 123-154. doi: 10.1007/978-3-540-75914-0_4.
[16]	F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. 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), 345-367. 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), 389-416. 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), 350-356, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533 Kluwer, Dordrecht, 1999.
[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.
[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.
[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.
[24]	A. S. Mischenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.
[25]	N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moskow Math. Soc., 26 (1972), 181-198.
[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.
[27]	M. Pavon, Hamilton-Jacobi equation for nonholonomic mechanics, J. Math. Phys., 46 (2005), 032902, 8pp. 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), 225-233. 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), 828-834. doi: 10.1007/BF01589294.
[30]	N. Woodhouse, Geometric Quantization, Oxford mathematical monographs. Oxford university press, second edition, 1991.

show all references

References:

[1]	R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, second edition, 1978.
[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.
[3]	V. I. Arnold and A. B. Givental, Symplectic Geometry, Dynamical systems IV, Encyclopaedia Math. Sci. Springer, 4 (2001), 1-138.
[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.
[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.
[6]	L. M. Bates, Examples of singular nonholonomic reduction, Rep. Math. Phys., 42 (1998), 231-247. 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), 29-35. 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), 295-308. doi: 10.1016/0034-4877(96)84069-9.
[9]	L. M. Bates and J. Śniatycki, Nonholonomic reduction, Rep. Math. Phys., 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N.
[10]	O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys., 196 (1998), 19-51. 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), 1417-1458. 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), 431-454. doi: 10.1142/S0219887810004385.
[13]	R. Cushman, D. Kemppeinen, J. Śniatycki and L. M. Bates, Geometry of nonholonomic constraints, Rep. Math. Phys., 36 (1995), 275-286. 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), 159-198. 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, Lecture Notes in Mathematics, 1927 (2008), 123-154. doi: 10.1007/978-3-540-75914-0_4.
[16]	F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Appl. Math., 87 (2005), 93-121. 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), 345-367. 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), 389-416. 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), 350-356, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533 Kluwer, Dordrecht, 1999.
[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.
[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.
[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.
[24]	A. S. Mischenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.
[25]	N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moskow Math. Soc., 26 (1972), 181-198.
[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.
[27]	M. Pavon, Hamilton-Jacobi equation for nonholonomic mechanics, J. Math. Phys., 46 (2005), 032902, 8pp. 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), 225-233. 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), 828-834. doi: 10.1007/BF01589294.
[30]	N. Woodhouse, Geometric Quantization, Oxford mathematical monographs. Oxford university press, second edition, 1991.

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