Variational reduction of Lagrangian systems with general constraints

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

Variational reduction of Lagrangian systems with general constraints

Sergio Grillo ^1, and Marcela Zuccalli ^2,

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

Abstract
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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.

Keywords: principal connections., Lagrangian systems, variational reduction.

Mathematics Subject Classification: Primary: 37J05, 70H30; Secondary: 70H5.

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.
[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.
[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.
[4]	W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," Second edition, Pure and Applied Mathematics, 120, Academic Press, Orlando, FL, 1986.
[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.
[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.
[7]	H. Cendra, S. Ferraro and S. Grillo, Lagrangian reduction of generalized nonholonomic systems, Journal of Geometry and Physics, 58 (2008), 1271-1290.
[8]	H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp.
[9]	H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 052904, 35 pp.
[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.
[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.
[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.
[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.
[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.
[15]	N. G. Chetaev, On Gauss principle, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68-71.
[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.
[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.
[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.
[19]	C. Godbillon, "Géométrie Différentielle et Mécanique Analytique," Hermann, Paris, 1969.
[20]	J. H. Greidanus, "Besturing en Stabiliteit van het Neuswielonderstel," Rapport V 1038, Nationaal Luchtvaartlaboratorium, Amsterdam, 1942.
[21]	S. Grillo, "Sistemas Noholónomos Generalizados," (Spanish), Ph.D thesis, Universidad Nacional del Sur, 2007.
[22]	S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 082901, 34 pp.
[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.
[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.
[25]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 1963.
[26]	O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.
[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.
[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.
[29]	J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 2001.
[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.
[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.
[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.
[33]	J. W. S. Rayleigh, "The Theory of Sound," Dover Publications, New York, 1945.
[34]	Do Shan, Equations of motion of systems with second-order nonlinear non-holonomic constraints, Prikl. Mat. Mekh., 37 (1973), 349-354.
[35]	W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians, Acta Physica Polonica B, 30 (1999), 2909-2978.
[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.
[37]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge University Press, Cambriedge, 1937.

show all references

References:

[1]	R. Abraham and J. E. Marsden, "Foundation of Mechanics," Benjaming Cummings, New York, 1985.
[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.
[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.
[4]	W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," Second edition, Pure and Applied Mathematics, 120, Academic Press, Orlando, FL, 1986.
[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.
[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.
[7]	H. Cendra, S. Ferraro and S. Grillo, Lagrangian reduction of generalized nonholonomic systems, Journal of Geometry and Physics, 58 (2008), 1271-1290.
[8]	H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 022902, 29 pp.
[9]	H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 052904, 35 pp.
[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.
[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.
[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.
[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.
[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.
[15]	N. G. Chetaev, On Gauss principle, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68-71.
[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.
[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.
[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.
[19]	C. Godbillon, "Géométrie Différentielle et Mécanique Analytique," Hermann, Paris, 1969.
[20]	J. H. Greidanus, "Besturing en Stabiliteit van het Neuswielonderstel," Rapport V 1038, Nationaal Luchtvaartlaboratorium, Amsterdam, 1942.
[21]	S. Grillo, "Sistemas Noholónomos Generalizados," (Spanish), Ph.D thesis, Universidad Nacional del Sur, 2007.
[22]	S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 082901, 34 pp.
[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.
[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.
[25]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 1963.
[26]	O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.
[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.
[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.
[29]	J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 2001.
[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.
[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.
[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.
[33]	J. W. S. Rayleigh, "The Theory of Sound," Dover Publications, New York, 1945.
[34]	Do Shan, Equations of motion of systems with second-order nonlinear non-holonomic constraints, Prikl. Mat. Mekh., 37 (1973), 349-354.
[35]	W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians, Acta Physica Polonica B, 30 (1999), 2909-2978.
[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.
[37]	E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," Cambridge University Press, Cambriedge, 1937.

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