Lie and Noether Symmetries of a Class of Time-Dependent Anharmonic Oscillators.

Abstract

Lie's method of extended groups of point transformations is applied to a class of time-dependent, nonlinear oscillators with cubic nonlinearity. A classification of different cases with respect to their Lie point symmetries is presented and the corresponding reductions of the order of each equation are performed. In some cases a second reduction, i.e. integration, is possible due to the special character of the symmetry, namely to preserve also the action integral. In these cases the corresponding general solution is analytically given in terms of the elliptic integral of the first kind.