The principle of symmetric criticality
 Richard S. Palais
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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 semisimple).
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 Title
 The principle of symmetric criticality
 Journal

Communications in Mathematical Physics
Volume 69, Issue 1 , pp 1930
 Cover Date
 19791001
 DOI
 10.1007/BF01941322
 Print ISSN
 00103616
 Online ISSN
 14320916
 Publisher
 SpringerVerlag
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 Authors

 Richard S. Palais ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Brandeis University, 02154, Waltham, Massachusetts, USA