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Reduction of invariant constrained systems using anholonomic frames
Clebsch optimal control formulation in mechanics
1. | Centre National de la Recherche Scientifique (CNRS), Laboratoire de Météorologie Dynamique, École Normale Supérieure, Paris, France |
2. | Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland |
References:
[1] |
A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,", Encyclopaedia of Mathematical Sciences, 87 ().
|
[2] |
V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier, 16 (1966), 319.
|
[3] |
A. M. Bloch, R. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces,, Commun. Math. Phys., 187 (1997), 357.
doi: 10.1007/s002200050140. |
[4] |
A. M. Bloch and P. E. Crouch, Optimal control and geodesic fows,, Syst. & Control Lett., 28 (1996), 65.
doi: 10.1016/0167-6911(96)00008-4. |
[5] |
A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow,, Proc. CDC IEEE, 39 (2000), 1273. Google Scholar |
[6] |
A. M. Bloch, P. E. Crouch, J. E. Marsden and T. S. Ratiu, Discrete rigid body dynamics and optimal control,, Proc. IEEE Conf. on Decision and Control, 37 (1998), 2249. Google Scholar |
[7] |
A. M. Bloch, P. E. Crouch and N. Nordkvist, Clebsch optimal control representation of mechanical systems and corresponding discretization,, preprint., (). Google Scholar |
[8] |
A. M. Bloch, P. E. Crouch, J. E. Marsden and A. K Sanyal, Optimal control and geodesics on quadratic matrix Lie groups,, Found. Comput. Math., 8 (2008), 469.
doi: 10.1007/s10208-008-9025-1. |
[9] |
A. M. Bloch, P. E. Crouch and A. K Sanyal, A variational problem on Stiefel manifolds,, Nonlinearity, 19 (2006), 2247.
doi: 10.1088/0951-7715/19/10/002. |
[10] |
R. Brockett, The double bracket equation as the solution of a variational problem,, Hamiltonian and gradient flows, 3 (1994), 69.
|
[11] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[12] |
C. J. Cotter, The variational particle-mesh method for matching curves,, J. Phys. A: Math. Theor., 41 (2008).
|
[13] |
C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles,, Found. Comput. Math., 9 (2009), 221.
doi: 10.1007/s10208-007-9022-9. |
[14] |
C. J. Cotter and D. D. Holm, Geodesic boundary value problems with symmetry,, J. Geom. Mech., 2 (2010), 51.
doi: 10.3934/jgm.2010.2.51. |
[15] |
C. J. Cotter, D. D. Holm and P. E. Hydon, Multisymplectic formulation of fluid dynamics using the inverse map,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2671.
|
[16] |
F. Gay-Balmaz, D. D. Holm, D. Meier, T. S. Ratiu and F.-X. Vialard [2011], Invariant higher-order variational problems,, Comm. Math. Phys., (). Google Scholar |
[17] |
F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,, J. Symplectic Geom., 6 (2008), 189.
|
[18] |
F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv. Appl. Math., 42 (2008), 176.
|
[19] |
F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems,, Physica D, 239 (2010), 1929.
doi: 10.1016/j.physd.2010.07.002. |
[20] |
F. Gay-Balmaz and C. Vizman [2011], Dual pairs in fluid dynamics,, Annals of Global Analysis and Geometry, (). Google Scholar |
[21] |
D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, In Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J. E. Marsden, (2002), 113.
|
[22] |
D. D. Holm, Euler's fluid equations: Optimal control vs optimization,, Phys. Lett. A, 373 (2009), 4354.
doi: 10.1016/j.physleta.2009.09.061. |
[23] |
D. D. Holm and B. A. Kupershmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity,, Physica D, 6 (1983), 347.
doi: 10.1016/0167-2789(83)90017-9. |
[24] |
D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation,, In The Breadth of Symplectic and Poisson Geometry, 232 (2004), 203.
|
[25] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. in Math., 137 (1998), 1.
doi: 10.1006/aima.1998.1721. |
[26] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, Phys. Rev. Lett., 349 (1998), 4173.
doi: 10.1103/PhysRevLett.80.4173. |
[27] |
M. Ise and M. Takeuchi, "Lie groups. I, II,", Translations of Mathematical Monographs, 85 (1991).
|
[28] |
O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds,, Geom. Dedicata, 81 (2000), 209.
doi: 10.1023/A:1005287907806. |
[29] |
S. V. Manakov, A remark on the integration of Euler',s equations of the dynamics of an n-dimensional rigid body,, Funct. Anal. Appl., 10 (1976), 328.
doi: 10.1007/BF01076037. |
[30] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).
|
[31] |
J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect product and reduction in mechanics,, Trans. Amer. Math. Soc., 281 (1984), 147.
doi: 10.1090/S0002-9947-1984-0719663-1. |
[32] |
J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Physica D, 7 (1983), 305.
doi: 10.1016/0167-2789(83)90134-3. |
[33] |
A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant,, Ann. Scient. Ec. Norm Sup., 18 (1985), 553.
|
[34] |
A. S. Mishchenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras,, Sov. Math. Dokl., 17 (1976), 1591. Google Scholar |
[35] |
A. S. Mishchenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras,, Sel. Math. Sov., 2 (1982), 207.
|
[36] |
T. Ratiu, The motion of the free $n$-dimensional rigid body,, Indiana U. Math J., 29 (1980), 609.
|
[37] |
L. Younes, "Shapes and Diffeomorphisms,", Applied Mathematical Sciences, 171 (2010).
|
show all references
References:
[1] |
A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,", Encyclopaedia of Mathematical Sciences, 87 ().
|
[2] |
V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier, 16 (1966), 319.
|
[3] |
A. M. Bloch, R. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces,, Commun. Math. Phys., 187 (1997), 357.
doi: 10.1007/s002200050140. |
[4] |
A. M. Bloch and P. E. Crouch, Optimal control and geodesic fows,, Syst. & Control Lett., 28 (1996), 65.
doi: 10.1016/0167-6911(96)00008-4. |
[5] |
A. M. Bloch, P. E. Crouch, D. D. Holm and J. E. Marsden, An optimal control formulation for inviscid incompressible ideal fluid flow,, Proc. CDC IEEE, 39 (2000), 1273. Google Scholar |
[6] |
A. M. Bloch, P. E. Crouch, J. E. Marsden and T. S. Ratiu, Discrete rigid body dynamics and optimal control,, Proc. IEEE Conf. on Decision and Control, 37 (1998), 2249. Google Scholar |
[7] |
A. M. Bloch, P. E. Crouch and N. Nordkvist, Clebsch optimal control representation of mechanical systems and corresponding discretization,, preprint., (). Google Scholar |
[8] |
A. M. Bloch, P. E. Crouch, J. E. Marsden and A. K Sanyal, Optimal control and geodesics on quadratic matrix Lie groups,, Found. Comput. Math., 8 (2008), 469.
doi: 10.1007/s10208-008-9025-1. |
[9] |
A. M. Bloch, P. E. Crouch and A. K Sanyal, A variational problem on Stiefel manifolds,, Nonlinearity, 19 (2006), 2247.
doi: 10.1088/0951-7715/19/10/002. |
[10] |
R. Brockett, The double bracket equation as the solution of a variational problem,, Hamiltonian and gradient flows, 3 (1994), 69.
|
[11] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[12] |
C. J. Cotter, The variational particle-mesh method for matching curves,, J. Phys. A: Math. Theor., 41 (2008).
|
[13] |
C. J. Cotter and D. D. Holm, Continuous and discrete Clebsch variational principles,, Found. Comput. Math., 9 (2009), 221.
doi: 10.1007/s10208-007-9022-9. |
[14] |
C. J. Cotter and D. D. Holm, Geodesic boundary value problems with symmetry,, J. Geom. Mech., 2 (2010), 51.
doi: 10.3934/jgm.2010.2.51. |
[15] |
C. J. Cotter, D. D. Holm and P. E. Hydon, Multisymplectic formulation of fluid dynamics using the inverse map,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2671.
|
[16] |
F. Gay-Balmaz, D. D. Holm, D. Meier, T. S. Ratiu and F.-X. Vialard [2011], Invariant higher-order variational problems,, Comm. Math. Phys., (). Google Scholar |
[17] |
F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,, J. Symplectic Geom., 6 (2008), 189.
|
[18] |
F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv. Appl. Math., 42 (2008), 176.
|
[19] |
F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems,, Physica D, 239 (2010), 1929.
doi: 10.1016/j.physd.2010.07.002. |
[20] |
F. Gay-Balmaz and C. Vizman [2011], Dual pairs in fluid dynamics,, Annals of Global Analysis and Geometry, (). Google Scholar |
[21] |
D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, In Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J. E. Marsden, (2002), 113.
|
[22] |
D. D. Holm, Euler's fluid equations: Optimal control vs optimization,, Phys. Lett. A, 373 (2009), 4354.
doi: 10.1016/j.physleta.2009.09.061. |
[23] |
D. D. Holm and B. A. Kupershmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity,, Physica D, 6 (1983), 347.
doi: 10.1016/0167-2789(83)90017-9. |
[24] |
D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation,, In The Breadth of Symplectic and Poisson Geometry, 232 (2004), 203.
|
[25] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. in Math., 137 (1998), 1.
doi: 10.1006/aima.1998.1721. |
[26] |
D. D. Holm, J. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion,, Phys. Rev. Lett., 349 (1998), 4173.
doi: 10.1103/PhysRevLett.80.4173. |
[27] |
M. Ise and M. Takeuchi, "Lie groups. I, II,", Translations of Mathematical Monographs, 85 (1991).
|
[28] |
O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds,, Geom. Dedicata, 81 (2000), 209.
doi: 10.1023/A:1005287907806. |
[29] |
S. V. Manakov, A remark on the integration of Euler',s equations of the dynamics of an n-dimensional rigid body,, Funct. Anal. Appl., 10 (1976), 328.
doi: 10.1007/BF01076037. |
[30] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).
|
[31] |
J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect product and reduction in mechanics,, Trans. Amer. Math. Soc., 281 (1984), 147.
doi: 10.1090/S0002-9947-1984-0719663-1. |
[32] |
J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Physica D, 7 (1983), 305.
doi: 10.1016/0167-2789(83)90134-3. |
[33] |
A. Medina and P. Revoy, Algèbres de Lie et produit scalaire invariant,, Ann. Scient. Ec. Norm Sup., 18 (1985), 553.
|
[34] |
A. S. Mishchenko and A. T. Fomenko, On the integration of the Euler equations on semisimple Lie algebras,, Sov. Math. Dokl., 17 (1976), 1591. Google Scholar |
[35] |
A. S. Mishchenko and A. T. Fomenko, Integrability of Euler equations on semisimple Lie algebras,, Sel. Math. Sov., 2 (1982), 207.
|
[36] |
T. Ratiu, The motion of the free $n$-dimensional rigid body,, Indiana U. Math J., 29 (1980), 609.
|
[37] |
L. Younes, "Shapes and Diffeomorphisms,", Applied Mathematical Sciences, 171 (2010).
|
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