December  2018, 10(4): 373-410. doi: 10.3934/jgm.2018014

Generalized variational calculus for continuous and discrete mechanical systems

1. 

Departamento de Matemática, Universidad Nacional del Sur (UNS), Avenida Alem 1253, 8000 Bahía Blanca, Argentina

2. 

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

Received  October 2014 Revised  September 2018 Published  November 2018

Fund Project: This work has been partially supported by UNS, Argentina (project PGI 24/ZL06); FONCYT, Argentina (project PICT 2010-2746); CONICET, Argentina (project PIP 2010–2012 11220090101018); MEC (Spain) Grants MTM2013-42870-P, MTM2009-08166-E, and IRSES-project "Geomech-246981".

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory... This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of skew-symmetric algebroids, therefore, our formalism easily extends to the case of Lie algebroids and nonholonomic systems (see also [20]). Hence, this framework automatically includes reduced mechanical systems subjected or not to constraints. Finally, we show that our formalism can be used to tackle the case of discrete mechanics, including reduced systems, systems subjected to constraints and discrete optimal control theory.

Citation: Viviana Alejandra Díaz, David Martín de Diego. Generalized variational calculus for continuous and discrete mechanical systems. Journal of Geometric Mechanics, 2018, 10 (4) : 373-410. doi: 10.3934/jgm.2018014
References:
[1]

R. Abraham and J. E. Marsden, Foundation of Mechanics, Addison Wesley, second edition, 1978.  Google Scholar

[2]

J. L. Anderson and P. G. Bergmann, Constraints in covariant field theories, Physical Rev. (2), 83 (1951), 1018-1025. doi: 10.1103/PhysRev.83.1018.  Google Scholar

[3]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, volume 3 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, third edition, 2006. [Dynamical systems. Ⅲ], Translated from the Russian original by E. Khukhro.  Google Scholar

[4]

M. Barbero-LiñánM. de LeónD. Martín de DiegoJ. C. Marrero and M. C. Muñoz Lecanda, Kinematic reduction and the Hamilton-Jacobi equation, Journal of Geometric Mechanics, 4 (2012), 207-237.  doi: 10.3934/jgm.2012.4.207.  Google Scholar

[5]

M. Barbero-Liñán and M. C. Muñoz-Lecanda, Geometric approach to Pontryagin's maximum principle, Acta Appl. Math., 108 (2009), 429-485.  doi: 10.1007/s10440-008-9320-5.  Google Scholar

[6]

A. M. BlochJ. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in non-holonomic systems, Dyn. Syst., 24 (2009), 187-222.  doi: 10.1080/14689360802609344.  Google Scholar

[7]

A. M. Bloch and P. E. Crouch, Nonholonomic and vakonomic control systems on Riemannian manifolds, Dynamics and Control of Mechanical Systems, the Falling Cat and Related Problems, 1 (1993), 25-52.   Google Scholar

[8]

B. Bonnard and M. Chyba, Sub-Riemannian geometry: The martinet case, In Geometric Control and Non-Holonomic Mechanics, volume 25, pages 79-100. Canadian Mathematical society, 1998.  Google Scholar

[9]

F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325.  doi: 10.1016/0393-0440(95)00016-X.  Google Scholar

[10]

J. Cortés MonforteM. de LeónD. Martín de Diego and S. Martinez, Geometric description of vakonomic and nonholonomic dynamics, comparison of solutions, SIAM Journal on Control and Optimization, 5 (2003), 1389-1412.  doi: 10.1137/S036301290036817X.  Google Scholar

[11]

J. Cortés MonforteM. de León and S. Martinez, The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems. the vakonomic bracket, J. Math. Phys., 41 (2000), 2090-2120.  doi: 10.1063/1.533229.  Google Scholar

[12]

M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics. Dynamical systems, An International Journal, 25 (2010), 159-187.  doi: 10.1080/14689360903360888.  Google Scholar

[13]

M. de León, F. Jiménez and D. Martín de Diego, Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings, J. Phys. A, 45 (2012), 205204, 29pp. doi: 10.1088/1751-8113/45/20/205204.  Google Scholar

[14]

M. de LeónJ. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. of Algebra, 129 (1990), 194-230.   Google Scholar

[15]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, volume 112 of North-Holland Mathematics Studies, North-Holland, Amsterdam, 1985).  Google Scholar

[16]

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

[17]

P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian J. Math., 2 (1950), 129-148.   Google Scholar

[18]

P. A. M. Dirac, Generalized Hamiltonian dynamics, Proc. Roy. Soc. London. Ser. A, 246 (1958), 326-332.   Google Scholar

[19]

P. A. M. Dirac, Lectures on Quantum Mechanics, volume 2 of Belfer Graduate School of Science Monographs Series, Belfer Graduate School of Science, New York; produced and distributed by Academic Press, Inc., New York, 1967. Second printing of the 1964 original.  Google Scholar

[20]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, Journal of Physics A: Mathematical and theoretical, 41 (2008), 175204, 25 pp. doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[21]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752.  Google Scholar

[22]

P. A. Griffiths, Exterior Differential Systems and the Calculus of Variations, volume 25 of Progress in Mathematics. Birkhäuser, Boston, Mass., 1983.  Google Scholar

[23]

D. D. Holm, Geometric Mechanics. Part Ⅰ, Imperial College Press, London, second edition, 2011. Dynamics and symmetry.  Google Scholar

[24]

D. D. Holm, Geometric Mechanics. Part Ⅱ. Rotating, Translating and Rolling, Imperial College Press, London, second edition, 2011. doi: 10.1142/p802.  Google Scholar

[25]

L. Hsu, Calculus of variations via the Griffiths formalism, J. Differential Geom., 36 (1992), 551-589.   Google Scholar

[26]

A. IbortM. de LeónJ. C. Marrero and D. Martín de Diego, Dirac brackets in constrained dynamics, Fortschr. Phys., 47 (1999), 459-492.  doi: 10.1002/(SICI)1521-3978(199906)47:5<459::AID-PROP459>3.0.CO;2-E.  Google Scholar

[27]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, A general framework for nonholonomic mechanics: Nonholonomic systems on Lie affgebroids, Journal of Mathematical Physics. Amer. Inst. Phys., 48 (2007), 083513, 45 pp. doi: 10.1063/1.2776845.  Google Scholar

[28]

D. IglesiasJ. C. MarreroD. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dynamical Systems: And International Journal, 23 (2008), 351-397.  doi: 10.1080/14689360802294220.  Google Scholar

[29]

D. Iglesias-PonteJ. C. MarreroD. Martín de Diego and E. Padrón, Discrete dynamics in implicit form, Discrete Contin. Dyn. Syst., 33 (2013), 1117-1135.  doi: 10.3934/dcds.2013.33.1117.  Google Scholar

[30]

V. V. Kozlov, On the realization of constraints in dynamics, Prikl. Mat. Mekh., 56 (1992), 692-698.  doi: 10.1016/0021-8928(92)90017-3.  Google Scholar

[31]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiments, Int. J. Nonlinear Mech., 30 (1995), 793-815.  doi: 10.1016/0020-7462(95)00024-0.  Google Scholar

[32]

F. L. Lewis, Optimal Control, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1986.  Google Scholar

[33]

K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, volume 124 of London Math. Soc. Lect. Notes Series, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[34]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.  Google Scholar

[35]

J. C. MarreroD. Martín de Diego and A. Stern, Symplectic groupoids and discrete constrained Lagrangian mechanics, DCDS-A, 35 (2015), 367-397.  doi: 10.3934/dcds.2015.35.367.  Google Scholar

[36]

J. C. MarreroD. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348.  doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[37]

J. C. Marrero, D. Martín de Diego and E. Martínez, The local description of discrete mechanics, Geometry, Mechanics, and Dynamics, 285-317, Fields Inst. Commun., 73, Springer, New York, 2015. doi: 10.1007/978-1-4939-2441-7_13.  Google Scholar

[38]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[39]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, volume 17. Springer-Verlag, New York, 1994. Second edition, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[40]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta. Appl. Math., 67 (2001), 295-320.  doi: 10.1023/A:1011965919259.  Google Scholar

[41]

E. Martínez, Reduction in optimal control theory, Rep. Math. Phys., 53 (2004), 79-90.  doi: 10.1016/S0034-4877(04)90005-5.  Google Scholar

[42]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, volume 91 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[43]

L. A. Pars, Treatise on Analytical Dynamics, Heinemann, London, 1965.  Google Scholar

[44]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated by D. E. Brown. A Pergamon Press Book. The Macmillan Co., New York, 1964.  Google Scholar

[45]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A675-A678.   Google Scholar

[46]

A. Weinstein, A universal phase space for particles in yang-mills fields, Lett. Math. Phys., 2 (1978), 417-420.  doi: 10.1007/BF00400169.  Google Scholar

[47]

A. Weinstein, Lagrangian mechanics and groupoids, Mechanics Day (Waterloo, ON, 1992) Fields Institute Communications, 7 (1996), 207-231.   Google Scholar

[48]

E. T. Whittaker, Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, 1959.  Google Scholar

[49]

G. Zampieri, Nonholonomic versus vakonomic dynamics, J. Diff. Equations, 163 (2000), 335-347.  doi: 10.1006/jdeq.1999.3727.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundation of Mechanics, Addison Wesley, second edition, 1978.  Google Scholar

[2]

J. L. Anderson and P. G. Bergmann, Constraints in covariant field theories, Physical Rev. (2), 83 (1951), 1018-1025. doi: 10.1103/PhysRev.83.1018.  Google Scholar

[3]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, volume 3 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, third edition, 2006. [Dynamical systems. Ⅲ], Translated from the Russian original by E. Khukhro.  Google Scholar

[4]

M. Barbero-LiñánM. de LeónD. Martín de DiegoJ. C. Marrero and M. C. Muñoz Lecanda, Kinematic reduction and the Hamilton-Jacobi equation, Journal of Geometric Mechanics, 4 (2012), 207-237.  doi: 10.3934/jgm.2012.4.207.  Google Scholar

[5]

M. Barbero-Liñán and M. C. Muñoz-Lecanda, Geometric approach to Pontryagin's maximum principle, Acta Appl. Math., 108 (2009), 429-485.  doi: 10.1007/s10440-008-9320-5.  Google Scholar

[6]

A. M. BlochJ. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in non-holonomic systems, Dyn. Syst., 24 (2009), 187-222.  doi: 10.1080/14689360802609344.  Google Scholar

[7]

A. M. Bloch and P. E. Crouch, Nonholonomic and vakonomic control systems on Riemannian manifolds, Dynamics and Control of Mechanical Systems, the Falling Cat and Related Problems, 1 (1993), 25-52.   Google Scholar

[8]

B. Bonnard and M. Chyba, Sub-Riemannian geometry: The martinet case, In Geometric Control and Non-Holonomic Mechanics, volume 25, pages 79-100. Canadian Mathematical society, 1998.  Google Scholar

[9]

F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325.  doi: 10.1016/0393-0440(95)00016-X.  Google Scholar

[10]

J. Cortés MonforteM. de LeónD. Martín de Diego and S. Martinez, Geometric description of vakonomic and nonholonomic dynamics, comparison of solutions, SIAM Journal on Control and Optimization, 5 (2003), 1389-1412.  doi: 10.1137/S036301290036817X.  Google Scholar

[11]

J. Cortés MonforteM. de León and S. Martinez, The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems. the vakonomic bracket, J. Math. Phys., 41 (2000), 2090-2120.  doi: 10.1063/1.533229.  Google Scholar

[12]

M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics. Dynamical systems, An International Journal, 25 (2010), 159-187.  doi: 10.1080/14689360903360888.  Google Scholar

[13]

M. de León, F. Jiménez and D. Martín de Diego, Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings, J. Phys. A, 45 (2012), 205204, 29pp. doi: 10.1088/1751-8113/45/20/205204.  Google Scholar

[14]

M. de LeónJ. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. of Algebra, 129 (1990), 194-230.   Google Scholar

[15]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, volume 112 of North-Holland Mathematics Studies, North-Holland, Amsterdam, 1985).  Google Scholar

[16]

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

[17]

P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian J. Math., 2 (1950), 129-148.   Google Scholar

[18]

P. A. M. Dirac, Generalized Hamiltonian dynamics, Proc. Roy. Soc. London. Ser. A, 246 (1958), 326-332.   Google Scholar

[19]

P. A. M. Dirac, Lectures on Quantum Mechanics, volume 2 of Belfer Graduate School of Science Monographs Series, Belfer Graduate School of Science, New York; produced and distributed by Academic Press, Inc., New York, 1967. Second printing of the 1964 original.  Google Scholar

[20]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, Journal of Physics A: Mathematical and theoretical, 41 (2008), 175204, 25 pp. doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[21]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752.  Google Scholar

[22]

P. A. Griffiths, Exterior Differential Systems and the Calculus of Variations, volume 25 of Progress in Mathematics. Birkhäuser, Boston, Mass., 1983.  Google Scholar

[23]

D. D. Holm, Geometric Mechanics. Part Ⅰ, Imperial College Press, London, second edition, 2011. Dynamics and symmetry.  Google Scholar

[24]

D. D. Holm, Geometric Mechanics. Part Ⅱ. Rotating, Translating and Rolling, Imperial College Press, London, second edition, 2011. doi: 10.1142/p802.  Google Scholar

[25]

L. Hsu, Calculus of variations via the Griffiths formalism, J. Differential Geom., 36 (1992), 551-589.   Google Scholar

[26]

A. IbortM. de LeónJ. C. Marrero and D. Martín de Diego, Dirac brackets in constrained dynamics, Fortschr. Phys., 47 (1999), 459-492.  doi: 10.1002/(SICI)1521-3978(199906)47:5<459::AID-PROP459>3.0.CO;2-E.  Google Scholar

[27]

D. Iglesias, J. C. Marrero, D. Martín de Diego and D. Sosa, A general framework for nonholonomic mechanics: Nonholonomic systems on Lie affgebroids, Journal of Mathematical Physics. Amer. Inst. Phys., 48 (2007), 083513, 45 pp. doi: 10.1063/1.2776845.  Google Scholar

[28]

D. IglesiasJ. C. MarreroD. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dynamical Systems: And International Journal, 23 (2008), 351-397.  doi: 10.1080/14689360802294220.  Google Scholar

[29]

D. Iglesias-PonteJ. C. MarreroD. Martín de Diego and E. Padrón, Discrete dynamics in implicit form, Discrete Contin. Dyn. Syst., 33 (2013), 1117-1135.  doi: 10.3934/dcds.2013.33.1117.  Google Scholar

[30]

V. V. Kozlov, On the realization of constraints in dynamics, Prikl. Mat. Mekh., 56 (1992), 692-698.  doi: 10.1016/0021-8928(92)90017-3.  Google Scholar

[31]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiments, Int. J. Nonlinear Mech., 30 (1995), 793-815.  doi: 10.1016/0020-7462(95)00024-0.  Google Scholar

[32]

F. L. Lewis, Optimal Control, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1986.  Google Scholar

[33]

K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, volume 124 of London Math. Soc. Lect. Notes Series, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[34]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.  Google Scholar

[35]

J. C. MarreroD. Martín de Diego and A. Stern, Symplectic groupoids and discrete constrained Lagrangian mechanics, DCDS-A, 35 (2015), 367-397.  doi: 10.3934/dcds.2015.35.367.  Google Scholar

[36]

J. C. MarreroD. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348.  doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[37]

J. C. Marrero, D. Martín de Diego and E. Martínez, The local description of discrete mechanics, Geometry, Mechanics, and Dynamics, 285-317, Fields Inst. Commun., 73, Springer, New York, 2015. doi: 10.1007/978-1-4939-2441-7_13.  Google Scholar

[38]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[39]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, volume 17. Springer-Verlag, New York, 1994. Second edition, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[40]

E. Martínez, Lagrangian mechanics on Lie algebroids, Acta. Appl. Math., 67 (2001), 295-320.  doi: 10.1023/A:1011965919259.  Google Scholar

[41]

E. Martínez, Reduction in optimal control theory, Rep. Math. Phys., 53 (2004), 79-90.  doi: 10.1016/S0034-4877(04)90005-5.  Google Scholar

[42]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, volume 91 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002.  Google Scholar

[43]

L. A. Pars, Treatise on Analytical Dynamics, Heinemann, London, 1965.  Google Scholar

[44]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated by D. E. Brown. A Pergamon Press Book. The Macmillan Co., New York, 1964.  Google Scholar

[45]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A675-A678.   Google Scholar

[46]

A. Weinstein, A universal phase space for particles in yang-mills fields, Lett. Math. Phys., 2 (1978), 417-420.  doi: 10.1007/BF00400169.  Google Scholar

[47]

A. Weinstein, Lagrangian mechanics and groupoids, Mechanics Day (Waterloo, ON, 1992) Fields Institute Communications, 7 (1996), 207-231.   Google Scholar

[48]

E. T. Whittaker, Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, 1959.  Google Scholar

[49]

G. Zampieri, Nonholonomic versus vakonomic dynamics, J. Diff. Equations, 163 (2000), 335-347.  doi: 10.1006/jdeq.1999.3727.  Google Scholar

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Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

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