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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 |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundation of Mechanics,", Benjaming Cummings, (1985). Google Scholar |
[2] |
P. Balseiro and J. Solomin, On generalized non-holonomic systems,, Letters of Mathematical Physics, 84 (2008), 15.
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.
doi: 10.1007/BF02199365. |
[4] |
W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry,", Second edition, 120 (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 (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.
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.
|
[8] |
H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006).
|
[9] |
H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).
|
[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.
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. 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.
|
[13] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages,, Memoirs of the American Mathematical Society, 152 (2001).
|
[14] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and non-holonomic systems,, in, (2001), 221.
|
[15] |
N. G. Chetaev, On Gauss principle,, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68. 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.
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 (1985).
|
[18] |
J. Fernandez, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems,, J. Geom. Mech., 2 (2010), 69.
doi: 10.3934/jgm.2010.2.69. |
[19] |
C. Godbillon, "Géométrie Différentielle et Mécanique Analytique,", Hermann, (1969).
|
[20] |
J. H. Greidanus, "Besturing en Stabiliteit van het Neuswielonderstel,", Rapport V 1038, (1038). Google Scholar |
[21] |
S. Grillo, "Sistemas Noholónomos Generalizados,", (Spanish), (2007). Google Scholar |
[22] |
S. Grillo, Higher order constrained Hamiltonian systems,, J. Math. Phys., 50 (2009).
|
[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.
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.
doi: 10.3934/jgm.2011.3.145. |
[25] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (1963). Google Scholar |
[26] |
O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.
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.
|
[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 (1994).
|
[29] |
J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications,", New York, (2001). Google Scholar |
[30] |
D. Pérez, "Sistemas Dinámicos no Holónomos Generalizados y su Aplicación a la Teoría de Control Automático Mediante Vínculos Cinemáticos,", (Spanish), (2006). Google Scholar |
[31] |
D. Pérez, "Sistemas Mecánicos con Vínculos de Orden Superior: Aplicaciones a la Teoría de Control,", (Spanish), (2007). Google Scholar |
[32] |
Y. Pironneau, Sur les liaisons non holonomes non linéaires déplacement virtuels à travail nul, conditions de Chetaev,, in, 117 (1983), 671.
|
[33] |
J. W. S. Rayleigh, "The Theory of Sound,", Dover Publications, (1945).
|
[34] |
Do Shan, Equations of motion of systems with second-order nonlinear non-holonomic constraints,, Prikl. Mat. Mekh., 37 (1973), 349. 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.
|
[36] |
V. Vâlcovici, Une extension des liaisons non holonomes et des principes variationnels,, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 102 (1958).
|
[37] |
E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1937).
|
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundation of Mechanics,", Benjaming Cummings, (1985). Google Scholar |
[2] |
P. Balseiro and J. Solomin, On generalized non-holonomic systems,, Letters of Mathematical Physics, 84 (2008), 15.
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.
doi: 10.1007/BF02199365. |
[4] |
W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry,", Second edition, 120 (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 (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.
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.
|
[8] |
H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006).
|
[9] |
H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).
|
[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.
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. 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.
|
[13] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages,, Memoirs of the American Mathematical Society, 152 (2001).
|
[14] |
H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and non-holonomic systems,, in, (2001), 221.
|
[15] |
N. G. Chetaev, On Gauss principle,, Izv. Fiz-Mat. Obsc. Kazan Univ., 7 (1934), 68. 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.
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 (1985).
|
[18] |
J. Fernandez, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems,, J. Geom. Mech., 2 (2010), 69.
doi: 10.3934/jgm.2010.2.69. |
[19] |
C. Godbillon, "Géométrie Différentielle et Mécanique Analytique,", Hermann, (1969).
|
[20] |
J. H. Greidanus, "Besturing en Stabiliteit van het Neuswielonderstel,", Rapport V 1038, (1038). Google Scholar |
[21] |
S. Grillo, "Sistemas Noholónomos Generalizados,", (Spanish), (2007). Google Scholar |
[22] |
S. Grillo, Higher order constrained Hamiltonian systems,, J. Math. Phys., 50 (2009).
|
[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.
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.
doi: 10.3934/jgm.2011.3.145. |
[25] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (1963). Google Scholar |
[26] |
O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.
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.
|
[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 (1994).
|
[29] |
J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications,", New York, (2001). Google Scholar |
[30] |
D. Pérez, "Sistemas Dinámicos no Holónomos Generalizados y su Aplicación a la Teoría de Control Automático Mediante Vínculos Cinemáticos,", (Spanish), (2006). Google Scholar |
[31] |
D. Pérez, "Sistemas Mecánicos con Vínculos de Orden Superior: Aplicaciones a la Teoría de Control,", (Spanish), (2007). Google Scholar |
[32] |
Y. Pironneau, Sur les liaisons non holonomes non linéaires déplacement virtuels à travail nul, conditions de Chetaev,, in, 117 (1983), 671.
|
[33] |
J. W. S. Rayleigh, "The Theory of Sound,", Dover Publications, (1945).
|
[34] |
Do Shan, Equations of motion of systems with second-order nonlinear non-holonomic constraints,, Prikl. Mat. Mekh., 37 (1973), 349. 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.
|
[36] |
V. Vâlcovici, Une extension des liaisons non holonomes et des principes variationnels,, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 102 (1958).
|
[37] |
E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1937).
|
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