2011, Vol.14, No.1, pp.49-59
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.
Key words:
Lie symmetries, Hamiltonian systems, nonlinear
time-dependent oscillators, solvability, Noether's theorem
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