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On the principle of stability of invariance of physical systems. (English) Zbl 0388.58022
MSC:
37C80 Symmetries, equivariant dynamical systems (MSC2010)
17B99 Lie algebras and Lie superalgebras
35L65 Hyperbolic conservation laws
37C75 Stability theory for smooth dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
81R40 Symmetry breaking in quantum theory
81T08 Constructive quantum field theory
70G99 General models, approaches, and methods in mechanics of particles and systems
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References:
[1] H. Poincaré , Les Méthodes Nouvelles de la Mécanique Céleste , Paris , 1899 . · JFM 30.0834.08
[2] R. Abraham and J. Marsden , Foundation of Mechanics , Benjamin , New York , 1967 .
[3] We do not list papers because of extreamly large numbers of papers in this field.
[4] R. Thom , Stabilité Structurelle et Morphogenèse , Benjamin , 1972 and reference there in. MR 488155 | Zbl 0294.92001 · Zbl 0294.92001
[5] H.D. Doebner , Nouvo Cim. , A 49 , 1967 , p. 306 ; Jour. of Math. Phys. , t. 9 , 1968 , p. 1638 and t. 11 , 1970 , p. 1463 . Zbl 0144.46201 · Zbl 0144.46201
[6] K.T. Shah , Topics in bifurcation theory , Seminar report, Clausthal, 1973 . · Zbl 0355.33006
[7] K.T. Shah , Reports on Math. Phys. , t. 6 , 1974 , p. 171 . MR 385018 | Zbl 0314.17011 · Zbl 0314.17011
[8] R. Thom , Symmetries gained and lost , Proceedings of III GIFT Seminars in Theor. Phys. , Madrid , 1972 ; see also L. Michel , Geometrical aspects of symmetry breaking, same proceedings . · Zbl 0352.58010
[9] R. Richardson , Jour. of Diff. Geom. , t. 3 , 1969 , p. 289 . Zbl 0215.38603 · Zbl 0215.38603
[10] R. Richardson , Proceedings of Symp. on Transformation Groups , Edited by P. Mostert , p. 429 , Springer-Verlag , 1967 . MR 244439
[11] M. Peixoto , ( Topology , t. 1 , 1962 , p. 101 , Ann. of Math. , t. 87 , 1968 , p. 422 ) has shown that if the dimension of the manifold is two, then the set of structurally stable vector field or dynamical system is a dense set on the set of all vector fields on this two dimensional manifold. In the case of differentiable maps i. e. C\infty -maps, one can define stability as follows. Let Mn and Np be the two C\infty -manifolds and let Cr (,) be the space of all maps from Mn to Np provided with the Cr-topology. A map is called stable if all’nearby maps’ k are of the same type topologically as f and the diagram is commutative, i. e. fh = h’k where h and h’ are \in -homeomorphisms of Mn and Np. For a general reference, see M. Golubitsky and Guillemin, Stable mappings and their Singularities , Springer-Verlag , 1973 . MR 142859
[12] E.P. Wigner and E. Inönü , Proc. Natl. Acad. Sci (USA), t. 39 , 1953 , p. 510 . MR 55352 | Zbl 0050.02601 · Zbl 0050.02601
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