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 \(\mathfrak {sl}(2,\mathbb {R})\oplus \mathfrak {so}(N)\).
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During the preparation of this work, the author was financially supported by the research project MTM2013-43820-P of the MINECO.
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Campoamor-Stursberg, R. Perturbations of Lagrangian systems based on the preservation of subalgebras of Noether symmetries. Acta Mech 227, 1941–1956 (2016). https://doi.org/10.1007/s00707-016-1621-6
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DOI: https://doi.org/10.1007/s00707-016-1621-6