March  2012, 4(1): 49-88. doi: 10.3934/jgm.2012.4.49

Variational reduction of Lagrangian systems with general constraints

1. 

Instituto Balseiro, U.N. de Cuyo - C.N.E.A., San Carlos de Bariloche, R8402AGP, Argentina

2. 

Departamento de Matemtica, Facultad de Ciencias Exactas, U.N.L.P., La Plata, Buenos Aires, Argentina

Received  October 2011 Revised  April 2012 Published  April 2012

In this paper we present an alternative procedure for reducing, in the Lagrangian formalism, the equations of motion of first order constrained mechanical systems with symmetry. The procedure involves two principal connections: one of them is used to define the reduced degrees of freedom and the other one to decompose variations into horizontal and vertical components. On the one hand, we show that this new procedure is particularly useful when the configuration space is a trivial principal bundle over the symmetry group, which is the case of many interesting examples. On the other hand, based on that procedure, we extend in a natural way the variational reduction methods to the Lagrangian systems with higher order constraints. Examples are discussed in order to illustrate the involved theorethical constructions.
Citation: Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49
References:
[1]

R. Abraham and J. E. Marsden, "Foundation of Mechanics," Benjaming Cummings, New York, 1985. Google Scholar

[2]

P. Balseiro and J. Solomin, On generalized non-holonomic systems, Letters of Mathematical Physics, 84 (2008), 15-30. doi: 10.1007/s11005-008-0236-9.  Google Scholar

[3]

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

[4]

W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," Second edition, Pure and Applied Mathematics, 120, Academic Press, Orlando, FL, 1986.  Google Scholar

[5]

F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.  Google Scholar

[6]

F. Cantrijn, M. de León, J. C. Marrero and D. Martín de Diego, Reduction of constrained systems with symmetries, J. Math. Phys., 40 (1999), 795-820. doi: 10.1063/1.532686.  Google Scholar

[7]

H. Cendra, S. Ferraro and S. Grillo, Lagrangian reduction of generalized nonholonomic systems, Journal of Geometry and Physics, 58 (2008), 1271-1290.  Google Scholar

[8]

H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp.  Google Scholar

[9]

H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 052904, 35 pp.  Google Scholar

[10]

H. Cendra, A. Ibort, M. de León and D. de Diego, A generalization of Chetaev's principle for a class of higher order nonholonomic constraints, J. Math. Phys., 45 (2004), 2785-2801. doi: 10.1063/1.1763245.  Google Scholar

[11]

H. Cendra, E. A. Lacomba and W. A. Reartes, The Lagrange D'Alembert-Poincaré equations for the symmetric rolling sphere, Actas del VI Congreso Antonio Monteiro, (2002), 19-32. Google Scholar

[12]

H. Cendra, J. E. Marsden, S. Pekarsky and T. S. Ratiu, Variational principles for Lie-Poisson and Hamilton-Poincaré equations, Moscow Mathematical Journal, 3 (2003), 833-867, 1197-1198.  Google Scholar

[13]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memoirs of the American Mathematical Society, 152 (2001), x+108 pp.  Google Scholar

[14]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and non-holonomic systems, in "Mathematics Unlimited-2001 and Beyond" (eds. B. Enguist and W. Schmid), Springer, Berlin, (2001), 221-273.  Google Scholar

[15]

N. G. Chetaev, On Gauss principle, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68-71. Google Scholar

[16]

M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Camb. Phil. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.  Google Scholar

[17]

M. de León and P. R. Rodrigues, "Generalized Classical Mechanics and Field Theory. A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives," North-Holland Mathematics Studies, 112, Notes on Pure Mathematics, 102, North-Holland Publishing Co., Amsterdam, 1985.  Google Scholar

[18]

J. Fernandez, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, J. Geom. Mech., 2 (2010), 69-111. doi: 10.3934/jgm.2010.2.69.  Google Scholar

[19]

C. Godbillon, "Géométrie Différentielle et Mécanique Analytique," Hermann, Paris, 1969.  Google Scholar

[20]

J. H. Greidanus, "Besturing en Stabiliteit van het Neuswielonderstel," Rapport V 1038, Nationaal Luchtvaartlaboratorium, Amsterdam, 1942. Google Scholar

[21]

S. Grillo, "Sistemas Noholónomos Generalizados," (Spanish), Ph.D thesis, Universidad Nacional del Sur, 2007. Google Scholar

[22]

S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 082901, 34 pp.  Google Scholar

[23]

S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, International Journal of Geometric Methods in Modern Physics, 7 (2010), 1-30. doi: 10.1142/S0219887810004580.  Google Scholar

[24]

S. Grillo, J. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization, Journal of Geometric Mechanics, 3 (2011), 145-196. doi: 10.3934/jgm.2011.3.145.  Google Scholar

[25]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 1963. Google Scholar

[26]

O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.  Google Scholar

[27]

C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364; Various approaches to conservative and nonconservative non-holonomic systems, Rep. Math. Phys., 42 (1998), 211-229, MR1656282.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar

[29]

J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 2001. Google Scholar

[30]

D. Pérez, "Sistemas Dinámicos no Holónomos Generalizados y su Aplicación a la Teoría de Control Automático Mediante Vínculos Cinemáticos," (Spanish), Proyecto Integrador, Carrera de Ingeniería Mecánica del Instituto Balseiro, 2006. Google Scholar

[31]

D. Pérez, "Sistemas Mecánicos con Vínculos de Orden Superior: Aplicaciones a la Teoría de Control," (Spanish), Maestría en Cs. Físicas del Instituto Balseiro, Orientación Física Aplicada, 2007. Google Scholar

[32]

Y. Pironneau, Sur les liaisons non holonomes non linéaires déplacement virtuels à travail nul, conditions de Chetaev, in "Proceedings of the IUTAM-ISIMMM Symposium on Modern Developments in Analytical Mechanics," Vol. II (Torino, 1982) (eds. S. Benenti, M. Francaviglia and A. Lichnerowicz), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 671-686.  Google Scholar

[33]

J. W. S. Rayleigh, "The Theory of Sound," Dover Publications, New York, 1945.  Google Scholar

[34]

Do Shan, Equations of motion of systems with second-order nonlinear non-holonomic constraints, Prikl. Mat. Mekh., 37 (1973), 349-354. Google Scholar

[35]

W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians, Acta Physica Polonica B, 30 (1999), 2909-2978.  Google Scholar

[36]

V. Vâlcovici, Une extension des liaisons non holonomes et des principes variationnels, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 102 (1958), 39 pp.  Google Scholar

[37]

E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge University Press, Cambriedge, 1937.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundation of Mechanics," Benjaming Cummings, New York, 1985. Google Scholar

[2]

P. Balseiro and J. Solomin, On generalized non-holonomic systems, Letters of Mathematical Physics, 84 (2008), 15-30. doi: 10.1007/s11005-008-0236-9.  Google Scholar

[3]

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

[4]

W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," Second edition, Pure and Applied Mathematics, 120, Academic Press, Orlando, FL, 1986.  Google Scholar

[5]

F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.  Google Scholar

[6]

F. Cantrijn, M. de León, J. C. Marrero and D. Martín de Diego, Reduction of constrained systems with symmetries, J. Math. Phys., 40 (1999), 795-820. doi: 10.1063/1.532686.  Google Scholar

[7]

H. Cendra, S. Ferraro and S. Grillo, Lagrangian reduction of generalized nonholonomic systems, Journal of Geometry and Physics, 58 (2008), 1271-1290.  Google Scholar

[8]

H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp.  Google Scholar

[9]

H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 052904, 35 pp.  Google Scholar

[10]

H. Cendra, A. Ibort, M. de León and D. de Diego, A generalization of Chetaev's principle for a class of higher order nonholonomic constraints, J. Math. Phys., 45 (2004), 2785-2801. doi: 10.1063/1.1763245.  Google Scholar

[11]

H. Cendra, E. A. Lacomba and W. A. Reartes, The Lagrange D'Alembert-Poincaré equations for the symmetric rolling sphere, Actas del VI Congreso Antonio Monteiro, (2002), 19-32. Google Scholar

[12]

H. Cendra, J. E. Marsden, S. Pekarsky and T. S. Ratiu, Variational principles for Lie-Poisson and Hamilton-Poincaré equations, Moscow Mathematical Journal, 3 (2003), 833-867, 1197-1198.  Google Scholar

[13]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Memoirs of the American Mathematical Society, 152 (2001), x+108 pp.  Google Scholar

[14]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and non-holonomic systems, in "Mathematics Unlimited-2001 and Beyond" (eds. B. Enguist and W. Schmid), Springer, Berlin, (2001), 221-273.  Google Scholar

[15]

N. G. Chetaev, On Gauss principle, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68-71. Google Scholar

[16]

M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Camb. Phil. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.  Google Scholar

[17]

M. de León and P. R. Rodrigues, "Generalized Classical Mechanics and Field Theory. A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives," North-Holland Mathematics Studies, 112, Notes on Pure Mathematics, 102, North-Holland Publishing Co., Amsterdam, 1985.  Google Scholar

[18]

J. Fernandez, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, J. Geom. Mech., 2 (2010), 69-111. doi: 10.3934/jgm.2010.2.69.  Google Scholar

[19]

C. Godbillon, "Géométrie Différentielle et Mécanique Analytique," Hermann, Paris, 1969.  Google Scholar

[20]

J. H. Greidanus, "Besturing en Stabiliteit van het Neuswielonderstel," Rapport V 1038, Nationaal Luchtvaartlaboratorium, Amsterdam, 1942. Google Scholar

[21]

S. Grillo, "Sistemas Noholónomos Generalizados," (Spanish), Ph.D thesis, Universidad Nacional del Sur, 2007. Google Scholar

[22]

S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 082901, 34 pp.  Google Scholar

[23]

S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, International Journal of Geometric Methods in Modern Physics, 7 (2010), 1-30. doi: 10.1142/S0219887810004580.  Google Scholar

[24]

S. Grillo, J. Marsden and S. Nair, Lyapunov constraints and global asymptotic stabilization, Journal of Geometric Mechanics, 3 (2011), 145-196. doi: 10.3934/jgm.2011.3.145.  Google Scholar

[25]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 1963. Google Scholar

[26]

O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.  Google Scholar

[27]

C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364; Various approaches to conservative and nonconservative non-holonomic systems, Rep. Math. Phys., 42 (1998), 211-229, MR1656282.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar

[29]

J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 2001. Google Scholar

[30]

D. Pérez, "Sistemas Dinámicos no Holónomos Generalizados y su Aplicación a la Teoría de Control Automático Mediante Vínculos Cinemáticos," (Spanish), Proyecto Integrador, Carrera de Ingeniería Mecánica del Instituto Balseiro, 2006. Google Scholar

[31]

D. Pérez, "Sistemas Mecánicos con Vínculos de Orden Superior: Aplicaciones a la Teoría de Control," (Spanish), Maestría en Cs. Físicas del Instituto Balseiro, Orientación Física Aplicada, 2007. Google Scholar

[32]

Y. Pironneau, Sur les liaisons non holonomes non linéaires déplacement virtuels à travail nul, conditions de Chetaev, in "Proceedings of the IUTAM-ISIMMM Symposium on Modern Developments in Analytical Mechanics," Vol. II (Torino, 1982) (eds. S. Benenti, M. Francaviglia and A. Lichnerowicz), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 671-686.  Google Scholar

[33]

J. W. S. Rayleigh, "The Theory of Sound," Dover Publications, New York, 1945.  Google Scholar

[34]

Do Shan, Equations of motion of systems with second-order nonlinear non-holonomic constraints, Prikl. Mat. Mekh., 37 (1973), 349-354. Google Scholar

[35]

W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians, Acta Physica Polonica B, 30 (1999), 2909-2978.  Google Scholar

[36]

V. Vâlcovici, Une extension des liaisons non holonomes et des principes variationnels, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 102 (1958), 39 pp.  Google Scholar

[37]

E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge University Press, Cambriedge, 1937.  Google Scholar

[1]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of discrete mechanical systems by stages. Journal of Geometric Mechanics, 2016, 8 (1) : 35-70. doi: 10.3934/jgm.2016.8.35

[2]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[3]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69

[4]

E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1

[5]

Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433

[6]

Leonardo J. Colombo, María Emma Eyrea Irazú, Eduardo García-Toraño Andrés. A note on Hybrid Routh reduction for time-dependent Lagrangian systems. Journal of Geometric Mechanics, 2020, 12 (2) : 309-321. doi: 10.3934/jgm.2020014

[7]

Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399

[8]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[9]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[10]

Anouar Bahrouni, Marek Izydorek, Joanna Janczewska. Subharmonic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1841-1850. doi: 10.3934/dcdss.2019121

[11]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

[12]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204

[13]

Francesca Alessio, Vittorio Coti Zelati, Piero Montecchiari. Chaotic behavior of rapidly oscillating Lagrangian systems. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 687-707. doi: 10.3934/dcds.2004.10.687

[14]

Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1

[15]

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

[16]

Mike Crampin, Tom Mestdag. Reduction of invariant constrained systems using anholonomic frames. Journal of Geometric Mechanics, 2011, 3 (1) : 23-40. doi: 10.3934/jgm.2011.3.23

[17]

Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317

[18]

Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137

[19]

Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74

[20]

Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]