Deformations of Lagrangian systems preserving a fixed subalgebra of Noether symmetries

Systems of second-order ordinary differential equations admitting a Lagrangian formulation are deformed requiring that the extended Lagrangian preserves a fixed subalgebra of Noether symmetries of the original system. For the case of the simple Lie algebra sl(2,R), this provides non-linear systems with two independent constants of the motion quadratic in the velocities. In the case of scalar differential equations, it is shown that equations of Pinney-type arise as the most general deformation of the time-dependent harmonic oscillator preserving a sl(2,R)subalgebra. The procedure is generalized naturally to two dimensions. In particular, it is shown that any deformation of the time-dependent harmonic oscillator in two dimensions that preserves a sl(2,R) subalgebra of Noether symmetries is equivalent to a generalized Ermakov-Ray-Reid system that satisfies the Helmholtz conditions of the Inverse Problem of Lagrangian Mechanics. Application of the procedure to other types of Lagrangians is illustrated. PACS numbers: 02.20Sv, 02.30Hq, 03.65Db, 45.20Jj