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This paper presents symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group.
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H. Cendra, D. D. Holm, J. E. Marsden, and T.S. Ratiu, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, Geometry of Differential Equations
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D. D. Holm, J. E. Marsden, and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum mechanics, Adv. Math
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J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems
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M. Castrillón López, T. S. Ratiu, and S. Shkoller, Reduction in principal fiber bundles: covariant Euler-Poincaré equations, Proc. Amer. Math. Soc
2000
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C. Kane, J. E. Marsden, M. Ortiz, and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Int. J. Numer. Methods Eng
2000
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M. Castrillón López, P. García Pérez, and T. S. Ratiu, Euler-Poincaré reduction on principal bundles, Lett. Math. Phys
2001
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H. Cendra, J. E. Marsden, and T.S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc
2001
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H. Cendra, J. E. Marsden, and T.S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems, in Mathematics Unlimited - 2001 and Beyond
2001
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J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer
2001
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D. D. Holm, J. E. Marsden, and T. S. Ratiu, The Euler-Poincaré equations in geophysical fluid dynamics, in Large-Scale Atmosphere-Ocean Dynamics
2002
Cited alongside, same era.
M. Castrillón López and T. S. Ratiu, Reduction in principal bundles: covariant Lagrange-Poincaré equations, Comm. Math. Phys
2003
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H. Cendra, J. E. Marsden, S, Pekarsky, and T.S. Ratiu, Variational principles for Lie-Poisson and Hamilton-Poincaré equations, Mosc. Math. J
2003
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R. C. Fetecau, J. E. Marsden, M. Ortiz, and M. West, Nonsmooth Lagrangian mechanics and variational collision integrators, SIAM J. Appl. Dyn. Syst
2003
Cited alongside, same era.
Y. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach
2003
Cited alongside, same era.
S. Leyendecker, S. Oberblöbaum, J. E. Marsden, and M. Ortiz, Discrete mechanics and optimal control for constrained systems, Optim. Control Appl. Methods
2010
Later among the works it cites.
M. Tao, H. Owhadi, and J. E. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging, Multiscale Model. Simul
2010
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M. Bruveris, F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu, The momentum map representation of images, J. Nonlinear Sci
2011
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D. Ellis, F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu, Lagrange-Poincaré field equations, J. Geom. Phys
2011
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F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu, Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions, Bull. Braz. Math. Soc. (N.S.)
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A. Lew, J. E. Marsden, M. Ortiz, and M. West, Variational time integrators, Int. J. Numer. Methods Eng
2004
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A. Lew, J. E. Marsden, M. Ortiz, and M. West, An overview of variational integrators, in Finite Element Methods: 1970Õs and Beyond, CIMNE
2004
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H. Brenner, Kinematics of volume transport, Physica A
2005
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H. Brenner, Navier-Stokes revisited, Physica A
2005
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F. Cipriano and A. B. Cruzeiro, Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus, Comm. Math. Phys
2007
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N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal
2008
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P. Constantin and G. Iyer, A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations, Comm. Pure Appl. Math
2008
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2011
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F. Gay-Balmaz and T. S. Ratiu, Geometry of nonabelian charged fluids, Dyn. Partial Differ. Equ
2011
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F. Gay-Balmaz and T. S. Ratiu, Clebsch optimal control formulation in mechanics, J. Geom. Mech
2011
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S. Leyendecker and S. Ober-Blöbaum, A variational approach to multirate integration for constrained systems, in ECCOMAS Thematic Conference: Multibody Dynamics: Computational Methods and Applications, Brussels, Belgium, 4-7 July 2011
2011
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S. Ober-Blöbaum, O. Junge, and J. E. Marsden, Discrete mechanics and optimal control: an analysis, Control Optim. Calc. Var
2011
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M. Arnaudon and A. B. Cruzeiro, Lagrangian Navier-Stokes diffusions on manifolds: variational principle and stability, Bull. Sci. Math
2012
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F. Gay-Balmaz, D. D. Holm, D. Meier, T. S. Ratiu, and F.-X. Vialard, Invariant higher-order variational problems, Comm. Math. Phys
2012
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F. Gay-Balmaz, D. D. Holm, D. Meier, T. S. Ratiu, and F.-X. Vialard, Invariant higher-order variational problems II, J. Nonlinear Sci
2012
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F. Gay-Balmaz, D. D. Holm, V. Putkaradze, and T. S. Ratiu, Exact geometric theory of dendronized polymer dynamics, Adv. in Appl. Math
2012
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F. Gay-Balmaz, J. E. Marsden, and T. S. Ratiu, Reduced variational formulations in free boundary continuum mechanics, J. Nonlinear Sci
2012
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F. Gay-Balmaz, T. S. Ratiu, and C. Tronci, Euler-Poincaré approaches to nematodynamics, Acta Appl. Math
2012
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T. Koide and T. Kodama, Navier-Stokes, Gross-Pitaevskii and generalized diffusion equations using the stochastic variational method, J. Phys. A
2012
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F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu, Geometric dynamics of optimization, Commun. Math. Sci
2013
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F. Gay-Balmaz, T. S. Ratiu, and C. Tronci, Equivalent theories of liquid crystal dynamics, Arch. Ration. Mech. Anal
2013
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M. Arnaudon, X. Chen and A. B. Cruzeiro, Stochastic Euler-Poincaré reduction, J. Math. Phys
2014
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F. Gay-Balmaz, M. Monastyrsky, and T. S. Ratiu, Lagrangian reductions and integrable systems in condensed matter, Comm. Math. Phys
2015
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S. Hochgerner and T. S. Ratiu, Geometry of non-holonomic diffusion, J. Eur. Math. Soc
2015
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D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. R. Soc. A
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D. Wang and H. Wang, Global existence of martingale solutions to the three-dimensional stochastic compressible Navier-Stokes equations, Diff. and Int. Equations
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D. Breit and M. Hofmanová, Stochastic Navier-Stokes equations for compressible fluids, Indiana Univ. Math. J
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