A classical problem in the calculus of variations is the investigation of critical points of functionals {\cal L} on normed spaces
V
. The present work addresses the question: Under what conditions on the functional {\cal L} and the underlying space
V
does {\cal L} have at most one critical point?
A sufficient condition for uniqueness is given: the presence of a "variational sub-symmetry", i.e., a one-parameter group
G
of transformations of
V
, which strictly reduces the values of {\cal L}. The "method of transformation groups" is applied to second-order elliptic boundary value problems on Riemannian manifolds. Further applications include problems of geometric analysis and elasticity.
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