March  2018, 10(1): 1-41. doi: 10.3934/jgm.2018001

Lagrange-d'alembert-poincaré equations by several stages

Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253,8000 Bahía Blanca, Argentina

Received  June 2014 Revised  June 2017 Published  December 2017

The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincaré equations by several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle by several stages, extending methods known for the case of one stage in the previous literature. The case of Euler's disk is described as an illustrative example.

Citation: Hernán Cendra, Viviana A. Díaz. Lagrange-d'alembert-poincaré equations by several stages. Journal of Geometric Mechanics, 2018, 10 (1) : 1-41. doi: 10.3934/jgm.2018001
References:
[1]

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

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, volume60 of Graduate Texts in Mathematics. Springer Verlag, second edition, 1989.  Google Scholar

[3]

L. BatesH. Graumann and C. MacDonnell, Examples of Gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308.  doi: 10.1016/0034-4877(96)84069-9.  Google Scholar

[4]

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

[5]

A. M. Bloch, Nonholonomic Mechanics and Control, volume24 of Interdisciplinary Applied Mathematics. Springer Verlag, 2003.  Google Scholar

[6]

A. M. BlochJ. E. KrishnaprasadJ. E. Marsden and R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. An., 136 (1996), 21-99.  doi: 10.1007/BF02199365.  Google Scholar

[7]

F. CantrijnM. deLeónJ. 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

[8]

J. J. Cariñena and M. F. Rañada, Lagrangian systems with constraints: A geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen., 26 (1993), 1335-1351.  doi: 10.1088/0305-4470/26/6/016.  Google Scholar

[9]

H. Cendra and V. A. Díaz, The Lagrange-d'Alembert-Poincaré equations and integrability for the Euler's disk, Regular and Chaotic Dynamics, 12 (2007), 56-67.  doi: 10.1134/S1560354707010054.  Google Scholar

[10]

H. CendraJ. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction and nonholonomic systems, Mathematics Unlimited and Beyond, Springer, (2001), 221-273.   Google Scholar

[11]

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

[12]

H. Cendra, J. E. Marsden, T. S. Ratiu and H. Yoshimura, Dirac-Weinstein reduction of Dirac anchored vector bundles, 2009, preprint. Google Scholar

[13]

J. CortésM. l deLeónJ.C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst., 24 (2009), 213-271.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[14]

M. deLeón and D. Martínde Diego, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.  doi: 10.1063/1.531571.  Google Scholar

[15]

M. deLeónJ.C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308.   Google Scholar

[16]

K. EhlersJ. KoillerR. Montgomery and P. M. Ríos, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, The Breath of symplectic and Poisson geometry. Prog. Math., 232 (2005), 75-120.   Google Scholar

[17]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179.   Google Scholar

[18]

D.D. HolmJ.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.   Google Scholar

[19]

W. S. Koon and J. E. Marsden, Optimal control for holonomic and nonholonomic mechanical systems with syummetry and Lagrangian reduction, SIAM J. Control Optim., 35 (1997), 901-929.   Google Scholar

[20]

Ch.M. Marle, Reduction of constrained mechanical systems and stability of relative equilibria, Commun. Math. Phys., 174 (1995), 295-318.   Google Scholar

[21]

Ch.M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys., 42 (1998), 211-229.   Google Scholar

[22]

J. E. Marsden, G. Misiolek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages Number 1913 in Hamiltonian Reduction by Stages. Springer, 2007.  Google Scholar

[23]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, volume17. Springer-Verlag, New York, 1994. Second edition, 1999.  Google Scholar

[24]

T. Mestdag, Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework, J. Phys. A, 38 (2005), 10157-10179.   Google Scholar

[25]

J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of the American Mathematical Society, Providence, Rhode Island, 1972.  Google Scholar

[26]

A. M. Vershik, Classical and Non-Classical Dynamics with Constraints, volume 1108 of Global Analysis-Studies and Applications I. Lecture Notes in Mathematics. Springer, 2002. Google Scholar

[27]

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

show all references

References:
[1]

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

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, volume60 of Graduate Texts in Mathematics. Springer Verlag, second edition, 1989.  Google Scholar

[3]

L. BatesH. Graumann and C. MacDonnell, Examples of Gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308.  doi: 10.1016/0034-4877(96)84069-9.  Google Scholar

[4]

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

[5]

A. M. Bloch, Nonholonomic Mechanics and Control, volume24 of Interdisciplinary Applied Mathematics. Springer Verlag, 2003.  Google Scholar

[6]

A. M. BlochJ. E. KrishnaprasadJ. E. Marsden and R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. An., 136 (1996), 21-99.  doi: 10.1007/BF02199365.  Google Scholar

[7]

F. CantrijnM. deLeónJ. 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

[8]

J. J. Cariñena and M. F. Rañada, Lagrangian systems with constraints: A geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen., 26 (1993), 1335-1351.  doi: 10.1088/0305-4470/26/6/016.  Google Scholar

[9]

H. Cendra and V. A. Díaz, The Lagrange-d'Alembert-Poincaré equations and integrability for the Euler's disk, Regular and Chaotic Dynamics, 12 (2007), 56-67.  doi: 10.1134/S1560354707010054.  Google Scholar

[10]

H. CendraJ. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction and nonholonomic systems, Mathematics Unlimited and Beyond, Springer, (2001), 221-273.   Google Scholar

[11]

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

[12]

H. Cendra, J. E. Marsden, T. S. Ratiu and H. Yoshimura, Dirac-Weinstein reduction of Dirac anchored vector bundles, 2009, preprint. Google Scholar

[13]

J. CortésM. l deLeónJ.C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst., 24 (2009), 213-271.  doi: 10.3934/dcds.2009.24.213.  Google Scholar

[14]

M. deLeón and D. Martínde Diego, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.  doi: 10.1063/1.531571.  Google Scholar

[15]

M. deLeónJ.C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A, 38 (2005), R241-R308.   Google Scholar

[16]

K. EhlersJ. KoillerR. Montgomery and P. M. Ríos, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, The Breath of symplectic and Poisson geometry. Prog. Math., 232 (2005), 75-120.   Google Scholar

[17]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179.   Google Scholar

[18]

D.D. HolmJ.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.   Google Scholar

[19]

W. S. Koon and J. E. Marsden, Optimal control for holonomic and nonholonomic mechanical systems with syummetry and Lagrangian reduction, SIAM J. Control Optim., 35 (1997), 901-929.   Google Scholar

[20]

Ch.M. Marle, Reduction of constrained mechanical systems and stability of relative equilibria, Commun. Math. Phys., 174 (1995), 295-318.   Google Scholar

[21]

Ch.M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys., 42 (1998), 211-229.   Google Scholar

[22]

J. E. Marsden, G. Misiolek, J. P. Ortega, M. Perlmutter and T. S. Ratiu, Hamiltonian Reduction by Stages Number 1913 in Hamiltonian Reduction by Stages. Springer, 2007.  Google Scholar

[23]

J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, volume17. Springer-Verlag, New York, 1994. Second edition, 1999.  Google Scholar

[24]

T. Mestdag, Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework, J. Phys. A, 38 (2005), 10157-10179.   Google Scholar

[25]

J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of the American Mathematical Society, Providence, Rhode Island, 1972.  Google Scholar

[26]

A. M. Vershik, Classical and Non-Classical Dynamics with Constraints, volume 1108 of Global Analysis-Studies and Applications I. Lecture Notes in Mathematics. Springer, 2002. Google Scholar

[27]

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

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