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Equivariant Morse theory for flows and an application to the $ N$-body problem


Author: Filomena Pacella
Journal: Trans. Amer. Math. Soc. 297 (1986), 41-52
MSC: Primary 58F25; Secondary 58E05, 58F40, 70F10
DOI: https://doi.org/10.1090/S0002-9947-1986-0849465-9
MathSciNet review: 849465
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, using Conley's index and equivariant cohomology, some Morse type inequalities are deduced for a flow equivariant with respect to the action of a compact topological group.

In the case of a gradient flow induced by a nondegenerate smooth function these inequalities coincide with those described by R. Bott.

The theory is applied to the study of the central configurations of $ N$-bodies.


References [Enhancements On Off] (What's this?)

  • [1] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London A 308 (1982), 523-615. MR 702806 (85k:14006)
  • [2] V. Benci, A geometrical index for the group $ {S^1}$ and some applications to the study of periodic solutions of ordinary differential equations, Comm. Pure Appl. Math. 34 (1981), 393-432. MR 615624 (82k:58040)
  • [3] V. Benci and F. Pacella, Morse theory for symmetric functionals on the sphere and an application to a bifurcation problem, Nonlinear Analysis: Theory, Methods, and Applications 9 (1985), 763-773. MR 799882 (86m:58036)
  • [4] R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982), 331-358. MR 663786 (84m:58026a)
  • [5] -, Nondegenerate critical manifolds, Ann. of Math. (2) 60 (1954), 248-261. MR 0064399 (16:276f)
  • [6] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [7] C. C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1978. MR 511133 (80c:58009)
  • [8] C. C. Conley and E. Zehnder, Morse type index theory for flows and periodic solutions for hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207-253. MR 733717 (86b:58021)
  • [9] E. R. Fadell and S. Husseini, Relative cohomological index theories (to appear).
  • [10] E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for hamiltonian systems, Invent. Math. 45 (1978), 139-174. MR 0478189 (57:17677)
  • [11] K. H. Hofmann and P. S. Mostert, Cohomology theories for compact abelian groups, Springer-Verlag, New York, 1973. MR 0372113 (51:8330)
  • [12] D. Husemoller, Fibre bundles, Springer-Verlag, New York, 1966.
  • [13] J. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N.J., 1963. MR 0163331 (29:634)
  • [14] J. I. Palmore, Classifying relative equilibria, Bull. Amer. Math. Soc. 81 (1975), 489-491. MR 0363076 (50:15514)
  • [15] F. Pacella, Central configurations of the $ N$-body problem via the equivariant Morse theory, Arch. Rational Mech. Anal. (to appear). MR 856309 (87k:70014)
  • [16] -, Morse theory for flows in the presence of a symmetry group, M. R. C. Rep. No. 2717, July 1984.
  • [17] J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, Berlin and New York, 1983. MR 688146 (84d:35002)
  • [18] M. Shub, Appendix to Smale's paper: Diagonals and relative equilibria (Proc. Nuffic Summer School), Lecture Notes in Math., vol. 197, Springer, Berlin and New York, 1971, pp. 199-201. MR 0278700 (43:4430)
  • [19] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [20] A. G Wasserman, Equivariant differential topology, Topology 8 (1969), 127-150. MR 0250324 (40:3563)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0849465-9
Keywords: Conley's index, group actions, Morse inequalities, $ N$-body problem
Article copyright: © Copyright 1986 American Mathematical Society

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