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March  2021, 13(1): 25-53. doi: 10.3934/jgm.2021001

Contact Hamiltonian and Lagrangian systems with nonholonomic constraints

1. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C\Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain

2. 

Universidad de Alcalá (UAH) Campus Universitario. Ctra. Madrid-Barcelona, Km. 33, 600. 28805 Alcal´a de Henares, Madrid, Spain

3. 

Real Academia de Ciencias Exactas, Fisicas y Naturales C/de Valverde 22, 28004 Madrid, Spain

Dedicated to Professor Tony Bloch on the occasion of his 65th birthday

Received  November 2019 Revised  October 2020 Published  March 2021 Early access  December 2020

In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstrained Hamiltonian vector field. Finally, we construct the nonholonomic bracket, which is an almost Jacobi bracket on the space of observables and provides the nonholonomic dynamics.

Citation: Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, Redwood City, CA, 1978. doi: 10.1090/chel/364.

[2]

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

[3]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.  doi: 10.1007/BF02199365.

[4]

A. V. Borisov and I. S. Mamaev, On the history of the development of the nonholonomic dynamics, Regul. Chaotic Dyn., 7 (2002), 43-47.  doi: 10.1070/RD2002v007n01ABEH000194.

[5]

A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Methods Mod. Phys., 16 (2019), 51pp. doi: 10.1142/S0219887819400036.

[6]

A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 12pp. doi: 10.3390/e19100535.

[7]

A. Bravetti, M. de León, J. C. Marrero and E. Padrón, Invariant measures for contact Hamiltonian systems: Symplectic sandwiches with contact bread, J. Phys. A: Math. Theoret., 53 (2020). doi: 10.1088/1751-8121/abbaaa.

[8]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999.

[9]

S. A. Chaplygin, Analysis of the Dynamics of Non-Holonomic Systems, Gostekhizdat, Mosow-Leningrad, 1949.

[10]

M. de León and D. M. de Diego, A constraint algorithm for singular Lagrangians subjected to nonholonomic constraints, J. Math. Phys., 38 (1997), 3055-3062.  doi: 10.1063/1.532051.

[11]

M. de León and D. M. de Diego, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.  doi: 10.1063/1.531571.

[12]

M. de León and D. M. de Diego, Solving non-holonomic Lagrangian dynamics in terms of almost product structures, Extracta Math., 11 (1996), 325-347. 

[13]

M. de León and M. Lainz Valcázar, Contact Hamiltonian systems, J. Math. Phys., 60 (2019), 18pp. doi: 10.1063/1.5096475.

[14]

M. de León and M. Lainz Valcázar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geom. Phys., 153 (2020), 13pp. doi: 10.1016/j.geomphys.2020.103651.

[15]

M. de León and M. Lainz Valcázar, Singular Lagrangians and precontact Hamiltonian systems, Int. J. Geom. Methods Mod. Phys., 16 (2019), 39pp. doi: 10.1142/S0219887819501585.

[16]

M. de LeónJ. C. Marrero and D. M. de Diego, Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A, 30 (1997), 1167-1190.  doi: 10.1088/0305-4470/30/4/018.

[17]

M. de León and P. R. Rodrigues, Higher-order mechanical systems with constraints, Internat. J. Theoret. Phys., 31 (1992), 1303-1313.  doi: 10.1007/BF00673930.

[18]

M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989.

[19]

M. A. de León, A historical review on nonholomic mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 106 (2012), 191-284.  doi: 10.1007/s13398-011-0046-2.

[20]

J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas and N. Román-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods Mod. Phys., 17 (2020), 27pp. doi: 10.1142/S0219887820500905.

[21]

F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems, J. Geom. Phys., 111 (2017), 169-193.  doi: 10.1016/j.geomphys.2016.08.018.

[22]

F. Gay-Balmaz and H. Yoshimura, From Lagrangian mechanics to nonequilibrium thermodynamics: A variational perspective, Entropy, 21 (2019). doi: 10.3390/e21010008.

[23]

B. Georgieva, The variational principle of Hergloz and related resultst, in Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2011, 214–225.

[24]

B. GeorgievaR. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927.  doi: 10.1063/1.1597419.

[25]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 2006.

[26]

G. Herglotz, Beruhrungstransformationen, in Lectures at the University of Gottingen, Gottingen, 1930.

[27]

A. Ibort, M. de León, G. Marmo and D. M. de Diego, Non-holonomic constrained systems as implicit differential equations. Geometrical structures for physical theories, I (Vietri, 1996), Rend. Sem. Mat. Univ. Politec. Torino, 54 (1996), 295–-317.

[28]

A. A. Kirillov, Local Lie algebras, Uspehi Mat. Nauk, 31 (1976), 57-76.  doi: 10.1070/RM1976v031n04ABEH001556.

[29]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.  doi: 10.1007/BF00375092.

[30]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regul. Chaotic Dyn., 7 (2002), 161-176.  doi: 10.1070/RD2002v007n02ABEH000203.

[31]

V. V. Kozlov, Realization of nonintegrable constraints in classical mechanics, Dokl. Akad. Nauk SSSR, 272 (1983), 550-554. 

[32]

A. V. Kremnev and A. S. Kuleshov, Nonlinear dynamics and stability of the skateboard, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 85-103.  doi: 10.3934/dcdss.2010.3.85.

[33]

A. S. Kuleshov, A mathematical model of the snakeboard, Mat. Model., 18 (2006), 37-48. 

[34]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[35]

A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. (9), 57 (1978), 453-488. 

[36]

Q. LiuP. J. Torres and C. Wang, Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Physics, 395 (2018), 26-44.  doi: 10.1016/j.aop.2018.04.035.

[37]

N. K. Moshchuk, On the motion of Chaplygin's sledge, J. Appl. Math. Mech., 51 (1987), 426-430.  doi: 10.1016/0021-8928(87)90079-7.

[38]

J. I. Ne${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$mark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33, American Mathematical Society, Providence, RI, 1972. doi: 10.1090/mmono/033.

[39]

V. V. Rumiantsev, On Hamilton's principle for nonholonomic systems, Prikl. Mat. Mekh., 42 (1978), 387-399. 

[40]

V. V. Rumyantsev, Variational principles for systems with unilateral constraints, J. Appl. Math. Mech., 70 (2006), 808-818.  doi: 10.1016/j.jappmathmech.2007.01.002.

[41]

A. A. Simoes, D. M. de Diego, M. de León and M. L. Valcázar, On the geometry of discrete contact mechanics, preprint, arXiv: 2003.11892.

[42]

A. A. SimoesM. de LeónM. L. Valcázar and D. M. de Diego, Contact geometry for simple thermodynamical systems with friction, Proc. A, 476 (2020), 244-259.  doi: 10.1098/rspa.2020.0244.

[43]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.

[44]

A. van der Schaft, Classical thermodynamics revisited: A systems and control perspective, preprint, arXiv: 2010.04213.

[45]

M. Vermeeren, A. Bravetti and M. Seri, Contact variational integrators, J. Phys. A, 52 (2019), 28pp. doi: 10.1088/1751-8121/ab4767.

[46]

A. M. Vershik and L. D. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Soviet Physics. Doklady, 17 (1972), 34-36. 

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea Publishing, Redwood City, CA, 1978. doi: 10.1090/chel/364.

[2]

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

[3]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.  doi: 10.1007/BF02199365.

[4]

A. V. Borisov and I. S. Mamaev, On the history of the development of the nonholonomic dynamics, Regul. Chaotic Dyn., 7 (2002), 43-47.  doi: 10.1070/RD2002v007n01ABEH000194.

[5]

A. Bravetti, Contact geometry and thermodynamics, Int. J. Geom. Methods Mod. Phys., 16 (2019), 51pp. doi: 10.1142/S0219887819400036.

[6]

A. Bravetti, Contact Hamiltonian dynamics: The concept and its use, Entropy, 19 (2017), 12pp. doi: 10.3390/e19100535.

[7]

A. Bravetti, M. de León, J. C. Marrero and E. Padrón, Invariant measures for contact Hamiltonian systems: Symplectic sandwiches with contact bread, J. Phys. A: Math. Theoret., 53 (2020). doi: 10.1088/1751-8121/abbaaa.

[8]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999.

[9]

S. A. Chaplygin, Analysis of the Dynamics of Non-Holonomic Systems, Gostekhizdat, Mosow-Leningrad, 1949.

[10]

M. de León and D. M. de Diego, A constraint algorithm for singular Lagrangians subjected to nonholonomic constraints, J. Math. Phys., 38 (1997), 3055-3062.  doi: 10.1063/1.532051.

[11]

M. de León and D. M. de Diego, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.  doi: 10.1063/1.531571.

[12]

M. de León and D. M. de Diego, Solving non-holonomic Lagrangian dynamics in terms of almost product structures, Extracta Math., 11 (1996), 325-347. 

[13]

M. de León and M. Lainz Valcázar, Contact Hamiltonian systems, J. Math. Phys., 60 (2019), 18pp. doi: 10.1063/1.5096475.

[14]

M. de León and M. Lainz Valcázar, Infinitesimal symmetries in contact Hamiltonian systems, J. Geom. Phys., 153 (2020), 13pp. doi: 10.1016/j.geomphys.2020.103651.

[15]

M. de León and M. Lainz Valcázar, Singular Lagrangians and precontact Hamiltonian systems, Int. J. Geom. Methods Mod. Phys., 16 (2019), 39pp. doi: 10.1142/S0219887819501585.

[16]

M. de LeónJ. C. Marrero and D. M. de Diego, Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A, 30 (1997), 1167-1190.  doi: 10.1088/0305-4470/30/4/018.

[17]

M. de León and P. R. Rodrigues, Higher-order mechanical systems with constraints, Internat. J. Theoret. Phys., 31 (1992), 1303-1313.  doi: 10.1007/BF00673930.

[18]

M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989.

[19]

M. A. de León, A historical review on nonholomic mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 106 (2012), 191-284.  doi: 10.1007/s13398-011-0046-2.

[20]

J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas and N. Román-Roy, New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries, Int. J. Geom. Methods Mod. Phys., 17 (2020), 27pp. doi: 10.1142/S0219887820500905.

[21]

F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems, J. Geom. Phys., 111 (2017), 169-193.  doi: 10.1016/j.geomphys.2016.08.018.

[22]

F. Gay-Balmaz and H. Yoshimura, From Lagrangian mechanics to nonequilibrium thermodynamics: A variational perspective, Entropy, 21 (2019). doi: 10.3390/e21010008.

[23]

B. Georgieva, The variational principle of Hergloz and related resultst, in Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2011, 214–225.

[24]

B. GeorgievaR. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem, J. Math. Phys., 44 (2003), 3911-3927.  doi: 10.1063/1.1597419.

[25]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 2006.

[26]

G. Herglotz, Beruhrungstransformationen, in Lectures at the University of Gottingen, Gottingen, 1930.

[27]

A. Ibort, M. de León, G. Marmo and D. M. de Diego, Non-holonomic constrained systems as implicit differential equations. Geometrical structures for physical theories, I (Vietri, 1996), Rend. Sem. Mat. Univ. Politec. Torino, 54 (1996), 295–-317.

[28]

A. A. Kirillov, Local Lie algebras, Uspehi Mat. Nauk, 31 (1976), 57-76.  doi: 10.1070/RM1976v031n04ABEH001556.

[29]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.  doi: 10.1007/BF00375092.

[30]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics, Regul. Chaotic Dyn., 7 (2002), 161-176.  doi: 10.1070/RD2002v007n02ABEH000203.

[31]

V. V. Kozlov, Realization of nonintegrable constraints in classical mechanics, Dokl. Akad. Nauk SSSR, 272 (1983), 550-554. 

[32]

A. V. Kremnev and A. S. Kuleshov, Nonlinear dynamics and stability of the skateboard, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 85-103.  doi: 10.3934/dcdss.2010.3.85.

[33]

A. S. Kuleshov, A mathematical model of the snakeboard, Mat. Model., 18 (2006), 37-48. 

[34]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[35]

A. Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. (9), 57 (1978), 453-488. 

[36]

Q. LiuP. J. Torres and C. Wang, Contact Hamiltonian dynamics: Variational principles, invariants, completeness and periodic behavior, Ann. Physics, 395 (2018), 26-44.  doi: 10.1016/j.aop.2018.04.035.

[37]

N. K. Moshchuk, On the motion of Chaplygin's sledge, J. Appl. Math. Mech., 51 (1987), 426-430.  doi: 10.1016/0021-8928(87)90079-7.

[38]

J. I. Ne${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$mark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33, American Mathematical Society, Providence, RI, 1972. doi: 10.1090/mmono/033.

[39]

V. V. Rumiantsev, On Hamilton's principle for nonholonomic systems, Prikl. Mat. Mekh., 42 (1978), 387-399. 

[40]

V. V. Rumyantsev, Variational principles for systems with unilateral constraints, J. Appl. Math. Mech., 70 (2006), 808-818.  doi: 10.1016/j.jappmathmech.2007.01.002.

[41]

A. A. Simoes, D. M. de Diego, M. de León and M. L. Valcázar, On the geometry of discrete contact mechanics, preprint, arXiv: 2003.11892.

[42]

A. A. SimoesM. de LeónM. L. Valcázar and D. M. de Diego, Contact geometry for simple thermodynamical systems with friction, Proc. A, 476 (2020), 244-259.  doi: 10.1098/rspa.2020.0244.

[43]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.

[44]

A. van der Schaft, Classical thermodynamics revisited: A systems and control perspective, preprint, arXiv: 2010.04213.

[45]

M. Vermeeren, A. Bravetti and M. Seri, Contact variational integrators, J. Phys. A, 52 (2019), 28pp. doi: 10.1088/1751-8121/ab4767.

[46]

A. M. Vershik and L. D. Faddeev, Differential geometry and Lagrangian mechanics with constraints, Soviet Physics. Doklady, 17 (1972), 34-36. 

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