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, 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.

[1]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461

[2]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[3]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295

[4]

Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317

[5]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

[6]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513

[7]

María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207

[8]

Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493

[9]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations and Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026

[10]

Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917

[11]

Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231

[12]

Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623

[13]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363

[14]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461

[15]

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

[16]

Gonzalo Dávila. Comparison principles for nonlocal Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022061

[17]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : i-iii. doi: 10.3934/dcdss.201805i

[18]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291

[19]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683

[20]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385

2021 Impact Factor: 0.737

Metrics

  • PDF downloads (149)
  • HTML views (0)
  • Cited by (5)

[Back to Top]