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March  2013, 5(1): 39-84. doi: 10.3934/jgm.2013.5.39

Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups

1. 

Laboratoire de Météorologie Dynamique, École Normale Supérieure/CNRS, F-75231 Paris, France

2. 

Department of Mathematics, University of Surrey, Guildford GU2 7XH

3. 

West University of Timişoara, RO-300223 Timişoara, Romania

Received  June 2012 Revised  January 2013 Published  April 2013

We formulate Euler-Poincaré equations on the Lie group $Aut(P)$ of automorphisms of a principal bundle $P$. The corresponding flows are referred to as EP$Aut$ flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle $P$, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing $\delta\text{-like}$ momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group $Aut_{vol}(P)$ of volume-preserving bundle automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.
Citation: François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39
References:
[1]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319. doi: 10.5802/aif.233.

[2]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", Benjamin-Cummings Publ. Co, (1985).

[3]

A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Kluwer Academic Publishers, (1997).

[4]

D. Bleecker, "Gauge Theory and Variational Principles,", Global Analysis Pure and Applied Series A, (1981).

[5]

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.

[6]

M. Chen, S. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions,, Lett. Math. Phys., 75 (2005), 1. doi: 10.1007/s11005-005-0041-7.

[7]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050.

[8]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102. doi: 10.2307/1970699.

[9]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence,, Phys. D, 152/153 (2001), 505. doi: 10.1016/S0167-2789(01)00191-9.

[10]

L. C. Garcia de Andrade, Vortex filaments in MHD,, Phys. Scr., 73 (2006), 484. doi: 10.1088/0031-8949/73/5/012.

[11]

F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations on manifolds with boundary,, Bull. Transilv. Univ. Braçsov Ser. III, 2 (2009), 55.

[12]

F. Gay-Balmaz and T. S. Ratiu, The Lie-Poisson structure of the LAE-$\alpha$ equation,, Dyn. Partial Differ. Equ., 2 (2005), 25.

[13]

F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,, J. Symplectic Geom., 6 (2008), 189.

[14]

F. Gay-Balmaz and T. S. Ratiu, Affine Lie-Poisson reduction, Yang-Mills magnetohydrodynamics, and superfluids,, J. Phys. A: Math. Theor., 41 (2008). doi: 10.1088/1751-8113/41/34/344007.

[15]

F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv. Appl. Math., 42 (2009), 176. doi: 10.1016/j.aam.2008.06.002.

[16]

F. Gay-Balmaz and T. S. Ratiu, Geometry of nonabelian charged fluids,, Dynamics of PDE, 8 (2011), 5.

[17]

F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group,, J. Math. Phys., 53 (2012).

[18]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics,, Ann. Glob. Anal. Geom., 41 (2011), 1. doi: 10.1007/s10455-011-9267-z.

[19]

F. Gay-Balmaz and C. Vizman, Dual pairs for nonabelian fluids,, preprint. 2013., (2013).

[20]

J. Gibbons, D. D. Holm and B. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics,, Phys. D, 6 (1983), 179. doi: 10.1016/0167-2789(83)90004-0.

[21]

S. Haller and C. Vizman, Non-linear Grassmannians as coadjoint orbits,, Math. Ann., 329 (2004), 771. doi: 10.1007/s00208-004-0536-z.

[22]

Y. Hattori, Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups,, J. Phys. A, 27 (1994). doi: 10.1088/0305-4470/27/2/004.

[23]

D. D. Holm, Hamiltonian structure for Alfven wave turbulence equations,, Phys. Lett. A, 108 (1985), 445. doi: 10.1016/0375-9601(85)90035-0.

[24]

D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, in, (2002), 113. doi: 10.1007/b97525.

[25]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation,, Progr. Math., 232 (2004), 203. doi: 10.1007/0-8176-4419-9_8.

[26]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721.

[27]

D. D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry,, Inverse Problems, 27 (0450). doi: 10.1088/0266-5611/27/4/045013.

[28]

D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids,, J. Math. Phys., 29 (1988), 21. doi: 10.1063/1.528176.

[29]

D. D. Holm, L. Ó Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.016601.

[30]

D. D. Holm, V. Putkaradze and S. N. Stechmann, Rotating concentric circular peakons,, Nonlinearity, 17 (2004), 2163. doi: 10.1088/0951-7715/17/6/008.

[31]

D. D. Holm, T. J. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy,, Neuroimage, 23 (2004).

[32]

D. D. Holm and C. Tronci, Geodesic flows on semidirect-product Lie groups: Geometry of singular measure-valued solutions,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2008), 457. doi: 10.1098/rspa.2008.0263.

[33]

D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures,, J. Geom. Mech., 2 (2009), 181. doi: 10.3934/jgm.2009.1.181.

[34]

R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms,, Moscow Math. J., 6 (2006), 307.

[35]

A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis,", Math. Surveys Monogr., 53 (1997).

[36]

B. Kostant, Quantization and unitary representations,, Lecture Notes in Math., 170 (1970), 87.

[37]

P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation,, Math. Notes, 81 (2007), 130. doi: 10.1134/S0001434607010142.

[38]

J. E. Marsden and P. J. Morrison, Noncanonical Hamiltonian field theory and reduced MHD,, Contemp. Math., 28 (1984), 133. doi: 10.1090/conm/028/751979.

[39]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).

[40]

J. E. Marsden, T. S. Ratiu and S. Shkoller, The geometry and analysis of the averaged Euler equations and a new diffeomorphism group,, Geom. Funct. Anal., 10 (2000), 582. doi: 10.1007/PL00001631.

[41]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Phys. D, 7 (1983), 305. doi: 10.1016/0167-2789(83)90134-3.

[42]

J. E. Marsden, A. Weinstein, T. S. Ratiu, R. Schmid and R. G. Spencer, Hamiltonian system with symmetry, coadjoint orbits and Plasma physics,, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 289.

[43]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", Second Edition, (1998).

[44]

R. Montogmery, J. E. Marsden and T. S. Ratiu, Gauged Lie-Poisson structures,, Contemp. Math., 28 (1984), 101. doi: 10.1090/conm/028/751976.

[45]

M. Molitor, The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations,, Differ. Geom. Appl., 28 (2010), 543. doi: 10.1016/j.difgeo.2010.04.005.

[46]

P. J. Morrison and R. D. Hazeltine, Hamiltonian formulation of reduced magnetohydrodynamics,, Phys. Fluids, 27 (1984), 886.

[47]

J.-P. Ortega and T. S. Ratiu, "Momentum maps and Hamiltonian reduction,", Progress in Mathematics 222, 222 (2004).

[48]

S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid,, J. Diff. Geom., 55 (2000), 145.

[49]

C. Vizman, Geodesics on extensions of Lie groups and stability: The superconductivity equation,, Phys. Lett. A, 284 (2001), 23. doi: 10.1016/S0375-9601(01)00279-1.

[50]

C. Vizman, Geodesic equations on diffeomorphism groups,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008). doi: 10.3842/SIGMA.2008.030.

[51]

C. Vizman, Natural differential forms on manifolds of functions,, Arch. Math. (Brno), 47 (2011), 201.

[52]

A. Weinstein, The local structure of Poisson manifolds,, J. Diff. Geom., 18 (1983), 523.

show all references

References:
[1]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319. doi: 10.5802/aif.233.

[2]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", Benjamin-Cummings Publ. Co, (1985).

[3]

A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Kluwer Academic Publishers, (1997).

[4]

D. Bleecker, "Gauge Theory and Variational Principles,", Global Analysis Pure and Applied Series A, (1981).

[5]

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.

[6]

M. Chen, S. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions,, Lett. Math. Phys., 75 (2005), 1. doi: 10.1007/s11005-005-0041-7.

[7]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050.

[8]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102. doi: 10.2307/1970699.

[9]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence,, Phys. D, 152/153 (2001), 505. doi: 10.1016/S0167-2789(01)00191-9.

[10]

L. C. Garcia de Andrade, Vortex filaments in MHD,, Phys. Scr., 73 (2006), 484. doi: 10.1088/0031-8949/73/5/012.

[11]

F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations on manifolds with boundary,, Bull. Transilv. Univ. Braçsov Ser. III, 2 (2009), 55.

[12]

F. Gay-Balmaz and T. S. Ratiu, The Lie-Poisson structure of the LAE-$\alpha$ equation,, Dyn. Partial Differ. Equ., 2 (2005), 25.

[13]

F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids,, J. Symplectic Geom., 6 (2008), 189.

[14]

F. Gay-Balmaz and T. S. Ratiu, Affine Lie-Poisson reduction, Yang-Mills magnetohydrodynamics, and superfluids,, J. Phys. A: Math. Theor., 41 (2008). doi: 10.1088/1751-8113/41/34/344007.

[15]

F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv. Appl. Math., 42 (2009), 176. doi: 10.1016/j.aam.2008.06.002.

[16]

F. Gay-Balmaz and T. S. Ratiu, Geometry of nonabelian charged fluids,, Dynamics of PDE, 8 (2011), 5.

[17]

F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group,, J. Math. Phys., 53 (2012).

[18]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics,, Ann. Glob. Anal. Geom., 41 (2011), 1. doi: 10.1007/s10455-011-9267-z.

[19]

F. Gay-Balmaz and C. Vizman, Dual pairs for nonabelian fluids,, preprint. 2013., (2013).

[20]

J. Gibbons, D. D. Holm and B. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics,, Phys. D, 6 (1983), 179. doi: 10.1016/0167-2789(83)90004-0.

[21]

S. Haller and C. Vizman, Non-linear Grassmannians as coadjoint orbits,, Math. Ann., 329 (2004), 771. doi: 10.1007/s00208-004-0536-z.

[22]

Y. Hattori, Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups,, J. Phys. A, 27 (1994). doi: 10.1088/0305-4470/27/2/004.

[23]

D. D. Holm, Hamiltonian structure for Alfven wave turbulence equations,, Phys. Lett. A, 108 (1985), 445. doi: 10.1016/0375-9601(85)90035-0.

[24]

D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, in, (2002), 113. doi: 10.1007/b97525.

[25]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation,, Progr. Math., 232 (2004), 203. doi: 10.1007/0-8176-4419-9_8.

[26]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721.

[27]

D. D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry,, Inverse Problems, 27 (0450). doi: 10.1088/0266-5611/27/4/045013.

[28]

D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids,, J. Math. Phys., 29 (1988), 21. doi: 10.1063/1.528176.

[29]

D. D. Holm, L. Ó Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.016601.

[30]

D. D. Holm, V. Putkaradze and S. N. Stechmann, Rotating concentric circular peakons,, Nonlinearity, 17 (2004), 2163. doi: 10.1088/0951-7715/17/6/008.

[31]

D. D. Holm, T. J. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy,, Neuroimage, 23 (2004).

[32]

D. D. Holm and C. Tronci, Geodesic flows on semidirect-product Lie groups: Geometry of singular measure-valued solutions,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2008), 457. doi: 10.1098/rspa.2008.0263.

[33]

D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures,, J. Geom. Mech., 2 (2009), 181. doi: 10.3934/jgm.2009.1.181.

[34]

R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms,, Moscow Math. J., 6 (2006), 307.

[35]

A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis,", Math. Surveys Monogr., 53 (1997).

[36]

B. Kostant, Quantization and unitary representations,, Lecture Notes in Math., 170 (1970), 87.

[37]

P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation,, Math. Notes, 81 (2007), 130. doi: 10.1134/S0001434607010142.

[38]

J. E. Marsden and P. J. Morrison, Noncanonical Hamiltonian field theory and reduced MHD,, Contemp. Math., 28 (1984), 133. doi: 10.1090/conm/028/751979.

[39]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Second Edition, (1999).

[40]

J. E. Marsden, T. S. Ratiu and S. Shkoller, The geometry and analysis of the averaged Euler equations and a new diffeomorphism group,, Geom. Funct. Anal., 10 (2000), 582. doi: 10.1007/PL00001631.

[41]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,, Phys. D, 7 (1983), 305. doi: 10.1016/0167-2789(83)90134-3.

[42]

J. E. Marsden, A. Weinstein, T. S. Ratiu, R. Schmid and R. G. Spencer, Hamiltonian system with symmetry, coadjoint orbits and Plasma physics,, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 289.

[43]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", Second Edition, (1998).

[44]

R. Montogmery, J. E. Marsden and T. S. Ratiu, Gauged Lie-Poisson structures,, Contemp. Math., 28 (1984), 101. doi: 10.1090/conm/028/751976.

[45]

M. Molitor, The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations,, Differ. Geom. Appl., 28 (2010), 543. doi: 10.1016/j.difgeo.2010.04.005.

[46]

P. J. Morrison and R. D. Hazeltine, Hamiltonian formulation of reduced magnetohydrodynamics,, Phys. Fluids, 27 (1984), 886.

[47]

J.-P. Ortega and T. S. Ratiu, "Momentum maps and Hamiltonian reduction,", Progress in Mathematics 222, 222 (2004).

[48]

S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid,, J. Diff. Geom., 55 (2000), 145.

[49]

C. Vizman, Geodesics on extensions of Lie groups and stability: The superconductivity equation,, Phys. Lett. A, 284 (2001), 23. doi: 10.1016/S0375-9601(01)00279-1.

[50]

C. Vizman, Geodesic equations on diffeomorphism groups,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008). doi: 10.3842/SIGMA.2008.030.

[51]

C. Vizman, Natural differential forms on manifolds of functions,, Arch. Math. (Brno), 47 (2011), 201.

[52]

A. Weinstein, The local structure of Poisson manifolds,, J. Diff. Geom., 18 (1983), 523.

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