SYMMETRY BOUNDS OF VARIATIONAL-PROBLEMS

Sharp upper bounds on the dimension of the Lie algebra of infinitesima l variational and divergence point symmetries of a non-trivial Lagrang ian L(x, u, u',..., u(n))(x, u is-an-element-of R) of arbitrary order n are found. For any given order, all Lagrangians whose Lie algebra of variational or of divergence symmetries is of maximal dimension are c ompletely classified, modulo local point transformations. It is shown, in particular, that for n greater-than-or-equal-to 2 the algebra of v ariational symmetries of the generalized free particle Lagrangian (u(n ))2 is not of maximal dimension, whereas when n = 1 there are several Lagrangians admitting a variational symmetry algebra of maximal dimens ion and generating differential equations different from the free part icle equation. A connection between variational problems on the line a nd scalar evolution equations in one time and one space variables is a lso established, showing that Lagrangians with a variational symmetry algebra of maximal dimension correspond to evolution equations with a maximal Lie algebra of time-preserving time-independent infinitesimal point symmetries. The technique used in the proof of the above results is applied to give a simple proof of the fact that an ordinary differ ential equation of order n > 2 has a symmetry algebra of maximal dimen sion if and only if it is locally equivalent under a point transformat ion to the generalized free particle equation u(n) = 0.