Abstract
We extend the classical theorem of L. Michel on the symmetry-breaking patterns in theories described by the minimization of a symmetric potential to the case of theories described by the minimization of a gauge-symmetric functional.
Similar content being viewed by others
References
Michel, L., Points critiques des fonctions invariants sur une G-variété,C. R. Acad. Sci. Paris A 272 (1971), 433.
Michel, L., Nonlinear group action. Smooth action of compact Lie groups on manifolds, in R. N. Sen and C. Weil (eds),Statistical Mechanics and Field Theory, Jerusalem, 1971.
Abud, M. and Sartori, G., The geometry of spontaneous symmetry breaking,Ann. Phys. 150, 307 (1983).
Michel, L. and Radicati, L., Properties of the breaking of hadronic internal symmetry,Ann. Phys. 66, 758 (1971); The geometry of the octet,Ann. Inst. Henri Poincaré 18, 185 (1973).
Palais, R. S., The principle of symmetric criticality,Comm. Math. Phys. 69, 19 (1979); Applications of the symmetric criticality principle in mathematical physics and differential geometry, in Gu Chaohao (ed.),Proc. 1981 Shanghai Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 1984.
Cicogna, G., Symmetry breakdown from bifurcation,Lett. Nuovo Cimento 31, 600 (1981).
Vanderbauwhede, A.,Local Bifurcation and Symmetry, Pitman, Boston, 1982.
Golubitsky, M. and Stewart, I., Hopf bifurcation in the presence of symmetry,Arch. Rational Mech. Anal. 87, 107 (1985).
Sattinger, D. H.,Branching in the Presence of Symmetry, S.I.A.M., Philadelphia, 1984.
Golubitsky, M., Schaeffer, D. and Stewart, I.,Singularities and Groups in Bifurcation Theory, Springer, New York, 1988.
Gaeta, G., Bifurcation and symmetry breaking,Phys. Rep. 189, 1 (1990).
Crawford, J. D. and Knobloch, E., Symmetry and symmetry-breaking bifurcations in fluid dynamics,Ann. Rev. Fluid Mech. 23, 341 (1991).
Gaeta, G., Bifurcations and nonlinear symmetries,Nonlinear Anal. 17 825 (1991).
Cicogna, G. and Gaeta, G., Lie-point symmetries in bifurcation problems,Ann. Inst. H. Poincaré 56, 375 (1992); Invariance of center manifold under Lie-Point symmetries, preprint C.P.Th. Ecole Polytechnique (1992), to appear inPhys Lett. A.
Cicogna, G., A nonlinear version of the equivariant bifurcation lemma,J. Phys. A 23, L1339 (1991).
Gaeta, G., Reduction and equivariant branching lemma: Dynamical systems, evolution PDEs, and gauge theories,Acta Appl. Math. 28 (1992), 43–68.
Olver, P. J.,Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
Bluman, G. W. and Kumei, S.,Symmetries and Differential Equations, Springer, Berlin, 1989.
Stephani, H.,Differential Equations. Their Solution Using Symmetries, Cambridge U.P., 1989.
Gaeta, G., Geometrical symmetries of nonlinear evolution equations and physics (volume in preparation); preprint C.P.Th. Ecole Polytechnique, 1992.
Gaeta, G., Michel theorem and critical orbits for gauge functionals, preprint C.P.Th. Ecole Polytechnique 1991; to be published inHelv. Phys. Acta.
Palais, R. S.,Mem. Amer. Math. Soc. 22 (1957).
Ehresmann, C., Les prolongements d'une variété differentiablé, I-V;C. R. Acad. Sci. Paris,233, 598 (1951);233, 777 (1951);233, 1081 (1951);234, 1028 (1952);234, 1424 (1952); see also234, 587 (1952);239, 1762 (1954);240, 397 (1955);240, 1755 (1955).
Arnold, V. I.,Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978, 1989;Geometrical Methods in the Theory of Ordinary Differential Equations; Springer, Berlin, 1983;Equations Differentielles Ordinaires, 2nd edn, Mir, Moscow, 1990.
Schwarz, G., Smooth functions invariant under a compact Lie group,Topology 14, 63 (1975). Lifting smooth homotopies of orbit spaces,Publ. Math. I.H.E.S. 51, 37 (1980).
Field, M. and Richardson, R., Symmetry breaking and the maximal isotropy subgroup conjecture for reflection groups,Arch. Rational Mech. Anal. 105, 61 (1989); Symmetry breaking in equivariant bifurcation problems,Bull. Amer. Math. Soc. 22, 79 (1990); Symmetry breaking and branching patterns in equivariant bifurcation theory,Arch. Rational Mech. Anal. 118, 297 (1992) and120, 147 (1990).