Perturbations of Lagrangian systems based on the preservation of subalgebras of Noether symmetries

Abstract

Systems of second-order ordinary differential equations admitting a Lagrangian formulation are perturbed requiring that the extended Lagrangian preserves a fixed subalgebra of Noether symmetries of the original system. For the simple Lie algebra $\mathfrak {sl}(2,\mathbb {R})$, this provides nonlinear systems with two independent constants of the motion quadratic in the velocities. Pinney-type equations are characterized as the most general $\mathfrak {sl}(2,\mathbb {R})$-preserving perturbation of the time-dependent (damped) harmonic oscillator. The procedure is generalized naturally to higher dimensions. In particular, it is shown that any perturbation of the time-dependent harmonic oscillator in two dimensions that preserves an $\mathfrak {sl}(2,\mathbb {R})$ 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 method to determine perturbations of the free Lagrangian in $\mathbb {R}^{N}$ is illustrated for the canonical chain of subalgebras of the Lie algebra $\mathfrak {sl}(2,\mathbb {R})\oplus \mathfrak {so}(N)$.