Communications in Mathematical Physics

, Volume 69, Issue 1, pp 19–30

The principle of symmetric criticality

  • Richard S. Palais


It is frequently explicitly or implicitly assumed that if a variational principle is invariant under some symmetry groupG, then to test whether a symmetric field configuration ϕ is an extremal, it suffices to check the vanishing of the first variation of the action corresponding to variations ϕ + δϕ that are also symmetric. We show by example that this is not valid in complete generality (and in certain cases its meaning may not even be clear), and on the other hand prove some theorems which validate its use under fairly general circumstances (in particular ifG is a group of Riemannian isometries, or if it is compact, or with some restrictions if it is semi-simple).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bochner, S.: Compact groups of differentiable transformations. Ann. Math.46, 372–381 (1945)Google Scholar
  2. 2.
    Coleman, S.: Classical lumps and their quantum descendants. Preprint, Harvard Physics Dept. (1975)Google Scholar
  3. 3.
    Eells, J., Lemaire, L.: A report on harmonic maps. Bull. London Math. Soc.10, 1–68 (1978)Google Scholar
  4. 4.
    Fuller, F.B.: Harmonic mappings. Proc. Nat. Acad. Sci. (U.S.A.)40, 110–116 (1954)Google Scholar
  5. 5.
    Guillemin, V., Sternberg, S.: Remarks on a paper of Hermann. Trans. Am. Math. Soc.130, 110–116 (1968)Google Scholar
  6. 6.
    Hermann, R.: The formal linearization of a semi-simple Lie algebra of vector fields about a singular point. Trans. Am. Math. Soc.130, 105–109 (1968)Google Scholar
  7. 7.
    Hsiang, W.Y.: On the compact, homogeneous, minimal submanifolds. Proc. Nat. Acad. Sci. (U.S.A.)50, 5–6 (1966)Google Scholar
  8. 8.
    Jacobson, N.: Lie algebras. New York: Interscience 1962Google Scholar
  9. 9.
    Lang, S.: Introduction to differentiable manifolds. New York: Interscience 1966Google Scholar
  10. 10.
    Misner, C.: Harmonic maps as models for physical theories. Phys. Rev. D (in press)Google Scholar
  11. 11.
    Palais, R.S.: Morse theory on Hilbert manifolds. Topology2, 299–340 (1963)Google Scholar
  12. 12.
    Palais, R.S.: Foundations of global nonlinear analysis. New York: Benjamin 1968Google Scholar
  13. 13.
    Pauli, W.: Theory of relativity. London: Pergamon Press 1958Google Scholar
  14. 14.
    Weyl, H.: Space-time-matter. New York: Dover 1951Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Richard S. Palais
    • 1
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA

Personalised recommendations