Uniqueness Theorems for Variational Problems by the Method of Transformation Groups 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