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

Contact Hamiltonian and Lagrangian systems with nonholonomic constraints

Manuel de León 1,3, , Víctor M. Jiménez 2, and  Manuel Lainz 1,

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.

Keywords: Nonholonomic constraints, contact Hamiltonian systems, Herglotz principle, dissipative systems, nonholonomic mechanics, Jacobi nonholonomic bracket.
Mathematics Subject Classification: 37J60;70F25;53D10;70H33.
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.  Google Scholar

[2]

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

[3]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.  doi: 10.1007/BF02199365.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

M. de León, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

B. Georgieva, R. 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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

[36]

Q. Liu, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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. Google Scholar

[42]

A. A. Simoes, M. de León, M. 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.  Google Scholar

[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.  Google Scholar

[44]

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

[45]

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

[46]

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

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.  Google Scholar

[2]

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

[3]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.  doi: 10.1007/BF02199365.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

M. de León, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

B. Georgieva, R. 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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

[36]

Q. Liu, P. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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. Google Scholar

[42]

A. A. Simoes, M. de León, M. 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.  Google Scholar

[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.  Google Scholar

[44]

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

[45]

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

[46]

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

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