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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), 319-361. doi: 10.5802/aif.233.  Google Scholar [2] R. Abraham and J. E. Marsden, "Foundations of Mechanics," Benjamin-Cummings Publ. Co, Updated 1985 version, reprinted by Perseus Publishing, second edition, 1978.  Google Scholar [3] A. Banyaga, "The Structure of Classical Diffeomorphism Groups," Kluwer Academic Publishers, 1997.  Google Scholar [4] D. Bleecker, "Gauge Theory and Variational Principles," Global Analysis Pure and Applied Series A, 1. Addison-Wesley Publishing Co., Reading, Mass, 1981.  Google Scholar [5] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar [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), 1-15. doi: 10.1007/s11005-005-0041-7.  Google Scholar [7] A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.  Google Scholar [8] D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163. doi: 10.2307/1970699.  Google Scholar [9] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.  Google Scholar [10] L. C. Garcia de Andrade, Vortex filaments in MHD, Phys. Scr., 73 (2006), 484-489. doi: 10.1088/0031-8949/73/5/012.  Google Scholar [11] F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations on manifolds with boundary, Bull. Transilv. Univ. Braçsov Ser. III, 2 (2009), 55-58.  Google Scholar [12] F. Gay-Balmaz and T. S. Ratiu, The Lie-Poisson structure of the LAE-$\alpha$ equation, Dyn. Partial Differ. Equ., 2 (2005), 25-57.  Google Scholar [13] F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.  Google Scholar [14] F. Gay-Balmaz and T. S. Ratiu, Affine Lie-Poisson reduction, Yang-Mills magnetohydrodynamics, and superfluids, J. Phys. A: Math. Theor., 41 (2008), 344007. doi: 10.1088/1751-8113/41/34/344007.  Google Scholar [15] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math., 42 (2009), 176-275. doi: 10.1016/j.aam.2008.06.002.  Google Scholar [16] F. Gay-Balmaz and T. S. Ratiu, Geometry of nonabelian charged fluids, Dynamics of PDE, 8 (2011), 5-19.  Google Scholar [17] F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys., 53 (2012), 123502. Google Scholar [18] F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Glob. Anal. Geom., 41 (2011), 1-24. doi: 10.1007/s10455-011-9267-z.  Google Scholar [19] F. Gay-Balmaz and C. Vizman, Dual pairs for nonabelian fluids, preprint. 2013. Google Scholar [20] J. Gibbons, D. D. Holm and B. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics, Phys. D, 6 (1983), 179-194. doi: 10.1016/0167-2789(83)90004-0.  Google Scholar [21] S. Haller and C. Vizman, Non-linear Grassmannians as coadjoint orbits, Math. Ann., 329 (2004), 771-785. doi: 10.1007/s00208-004-0536-z.  Google Scholar [22] Y. Hattori, Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups, J. Phys. A, 27 (1994), L21-L25. doi: 10.1088/0305-4470/27/2/004.  Google Scholar [23] D. D. Holm, Hamiltonian structure for Alfven wave turbulence equations, Phys. Lett. A, 108 (1985), 445-447. doi: 10.1016/0375-9601(85)90035-0.  Google Scholar [24] D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, in "Geometry, mechanics, and dynamics. Volume in honor of the 60th birthday of J. E. Marsden" (Edited by P. Newton et al.), New York, Springer, (2002), 113-167. doi: 10.1007/b97525.  Google Scholar [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), 203-235. doi: 10.1007/0-8176-4419-9_8.  Google Scholar [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), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar [27] D. D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry, Inverse Problems, 27 045013. doi: 10.1088/0266-5611/27/4/045013.  Google Scholar [28] D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids, J. Math. Phys., 29 (1988), 21-30. doi: 10.1063/1.528176.  Google Scholar [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), 016601. doi: 10.1103/PhysRevE.79.016601.  Google Scholar [30] D. D. Holm, V. Putkaradze and S. N. Stechmann, Rotating concentric circular peakons, Nonlinearity, 17 (2004), 2163-2186. doi: 10.1088/0951-7715/17/6/008.  Google Scholar [31] D. D. Holm, T. J. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy, Neuroimage, 23 (2004), S170-S178. Google Scholar [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), 457-476. doi: 10.1098/rspa.2008.0263.  Google Scholar [33] D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 2 (2009), 181-208. doi: 10.3934/jgm.2009.1.181.  Google Scholar [34] R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms, Moscow Math. J., 6 (2006), 307-315.  Google Scholar [35] A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis," Math. Surveys Monogr., 53, American Mathematical Society, Providence, RI. 1997.  Google Scholar [36] B. Kostant, Quantization and unitary representations, Lecture Notes in Math., 170 (1970), 87-208.  Google Scholar [37] P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation, Math. Notes, 81 (2007), 130-134. doi: 10.1134/S0001434607010142.  Google Scholar [38] J. E. Marsden and P. J. Morrison, Noncanonical Hamiltonian field theory and reduced MHD, Contemp. Math., 28 (1984), 133-150 doi: 10.1090/conm/028/751979.  Google Scholar [39] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Second Edition, Springer, 1999.  Google Scholar [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), 582-599. doi: 10.1007/PL00001631.  Google Scholar [41] J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323. doi: 10.1016/0167-2789(83)90134-3.  Google Scholar [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), 289-340.  Google Scholar [43] D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second Edition, Oxford Math. Monogr., Oxford University Press, 1998.  Google Scholar [44] R. Montogmery, J. E. Marsden and T. S. Ratiu, Gauged Lie-Poisson structures, Contemp. Math., 28 (1984), 101-114. doi: 10.1090/conm/028/751976.  Google Scholar [45] M. Molitor, The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations, Differ. Geom. Appl., 28 (2010), 543-564. doi: 10.1016/j.difgeo.2010.04.005.  Google Scholar [46] P. J. Morrison and R. D. Hazeltine, Hamiltonian formulation of reduced magnetohydrodynamics, Phys. Fluids, 27 (1984), 886-897. Google Scholar [47] J.-P. Ortega and T. S. Ratiu, "Momentum maps and Hamiltonian reduction," Progress in Mathematics 222, Boston Birkhäuser, 2004. Google Scholar [48] S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Diff. Geom., 55 (2000), 145-191.  Google Scholar [49] C. Vizman, Geodesics on extensions of Lie groups and stability: The superconductivity equation, Phys. Lett. A, 284 (2001), 23-30. doi: 10.1016/S0375-9601(01)00279-1.  Google Scholar [50] C. Vizman, Geodesic equations on diffeomorphism groups, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008). doi: 10.3842/SIGMA.2008.030.  Google Scholar [51] C. Vizman, Natural differential forms on manifolds of functions, Arch. Math. (Brno), 47 (2011), 201-215.  Google Scholar [52] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.  Google Scholar

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), 319-361. doi: 10.5802/aif.233.  Google Scholar [2] R. Abraham and J. E. Marsden, "Foundations of Mechanics," Benjamin-Cummings Publ. Co, Updated 1985 version, reprinted by Perseus Publishing, second edition, 1978.  Google Scholar [3] A. Banyaga, "The Structure of Classical Diffeomorphism Groups," Kluwer Academic Publishers, 1997.  Google Scholar [4] D. Bleecker, "Gauge Theory and Variational Principles," Global Analysis Pure and Applied Series A, 1. Addison-Wesley Publishing Co., Reading, Mass, 1981.  Google Scholar [5] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar [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), 1-15. doi: 10.1007/s11005-005-0041-7.  Google Scholar [7] A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.  Google Scholar [8] D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163. doi: 10.2307/1970699.  Google Scholar [9] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.  Google Scholar [10] L. C. Garcia de Andrade, Vortex filaments in MHD, Phys. Scr., 73 (2006), 484-489. doi: 10.1088/0031-8949/73/5/012.  Google Scholar [11] F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations on manifolds with boundary, Bull. Transilv. Univ. Braçsov Ser. III, 2 (2009), 55-58.  Google Scholar [12] F. Gay-Balmaz and T. S. Ratiu, The Lie-Poisson structure of the LAE-$\alpha$ equation, Dyn. Partial Differ. Equ., 2 (2005), 25-57.  Google Scholar [13] F. Gay-Balmaz and T. S. Ratiu, Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.  Google Scholar [14] F. Gay-Balmaz and T. S. Ratiu, Affine Lie-Poisson reduction, Yang-Mills magnetohydrodynamics, and superfluids, J. Phys. A: Math. Theor., 41 (2008), 344007. doi: 10.1088/1751-8113/41/34/344007.  Google Scholar [15] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv. Appl. Math., 42 (2009), 176-275. doi: 10.1016/j.aam.2008.06.002.  Google Scholar [16] F. Gay-Balmaz and T. S. Ratiu, Geometry of nonabelian charged fluids, Dynamics of PDE, 8 (2011), 5-19.  Google Scholar [17] F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys., 53 (2012), 123502. Google Scholar [18] F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Glob. Anal. Geom., 41 (2011), 1-24. doi: 10.1007/s10455-011-9267-z.  Google Scholar [19] F. Gay-Balmaz and C. Vizman, Dual pairs for nonabelian fluids, preprint. 2013. Google Scholar [20] J. Gibbons, D. D. Holm and B. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics, Phys. D, 6 (1983), 179-194. doi: 10.1016/0167-2789(83)90004-0.  Google Scholar [21] S. Haller and C. Vizman, Non-linear Grassmannians as coadjoint orbits, Math. Ann., 329 (2004), 771-785. doi: 10.1007/s00208-004-0536-z.  Google Scholar [22] Y. Hattori, Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups, J. Phys. A, 27 (1994), L21-L25. doi: 10.1088/0305-4470/27/2/004.  Google Scholar [23] D. D. Holm, Hamiltonian structure for Alfven wave turbulence equations, Phys. Lett. A, 108 (1985), 445-447. doi: 10.1016/0375-9601(85)90035-0.  Google Scholar [24] D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, in "Geometry, mechanics, and dynamics. Volume in honor of the 60th birthday of J. E. Marsden" (Edited by P. Newton et al.), New York, Springer, (2002), 113-167. doi: 10.1007/b97525.  Google Scholar [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), 203-235. doi: 10.1007/0-8176-4419-9_8.  Google Scholar [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), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar [27] D. D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry, Inverse Problems, 27 045013. doi: 10.1088/0266-5611/27/4/045013.  Google Scholar [28] D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids, J. Math. Phys., 29 (1988), 21-30. doi: 10.1063/1.528176.  Google Scholar [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), 016601. doi: 10.1103/PhysRevE.79.016601.  Google Scholar [30] D. D. Holm, V. Putkaradze and S. N. Stechmann, Rotating concentric circular peakons, Nonlinearity, 17 (2004), 2163-2186. doi: 10.1088/0951-7715/17/6/008.  Google Scholar [31] D. D. Holm, T. J. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy, Neuroimage, 23 (2004), S170-S178. Google Scholar [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), 457-476. doi: 10.1098/rspa.2008.0263.  Google Scholar [33] D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 2 (2009), 181-208. doi: 10.3934/jgm.2009.1.181.  Google Scholar [34] R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms, Moscow Math. J., 6 (2006), 307-315.  Google Scholar [35] A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis," Math. Surveys Monogr., 53, American Mathematical Society, Providence, RI. 1997.  Google Scholar [36] B. Kostant, Quantization and unitary representations, Lecture Notes in Math., 170 (1970), 87-208.  Google Scholar [37] P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation, Math. Notes, 81 (2007), 130-134. doi: 10.1134/S0001434607010142.  Google Scholar [38] J. E. Marsden and P. J. Morrison, Noncanonical Hamiltonian field theory and reduced MHD, Contemp. Math., 28 (1984), 133-150 doi: 10.1090/conm/028/751979.  Google Scholar [39] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Second Edition, Springer, 1999.  Google Scholar [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), 582-599. doi: 10.1007/PL00001631.  Google Scholar [41] J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D, 7 (1983), 305-323. doi: 10.1016/0167-2789(83)90134-3.  Google Scholar [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), 289-340.  Google Scholar [43] D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second Edition, Oxford Math. Monogr., Oxford University Press, 1998.  Google Scholar [44] R. Montogmery, J. E. Marsden and T. S. Ratiu, Gauged Lie-Poisson structures, Contemp. Math., 28 (1984), 101-114. doi: 10.1090/conm/028/751976.  Google Scholar [45] M. Molitor, The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations, Differ. Geom. Appl., 28 (2010), 543-564. doi: 10.1016/j.difgeo.2010.04.005.  Google Scholar [46] P. J. Morrison and R. D. Hazeltine, Hamiltonian formulation of reduced magnetohydrodynamics, Phys. Fluids, 27 (1984), 886-897. Google Scholar [47] J.-P. Ortega and T. S. Ratiu, "Momentum maps and Hamiltonian reduction," Progress in Mathematics 222, Boston Birkhäuser, 2004. Google Scholar [48] S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Diff. Geom., 55 (2000), 145-191.  Google Scholar [49] C. Vizman, Geodesics on extensions of Lie groups and stability: The superconductivity equation, Phys. Lett. A, 284 (2001), 23-30. doi: 10.1016/S0375-9601(01)00279-1.  Google Scholar [50] C. Vizman, Geodesic equations on diffeomorphism groups, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008). doi: 10.3842/SIGMA.2008.030.  Google Scholar [51] C. Vizman, Natural differential forms on manifolds of functions, Arch. Math. (Brno), 47 (2011), 201-215.  Google Scholar [52] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.  Google Scholar
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