R. Campoamor-Stursberg
Deformations of Lagrangian systems preserving a fixed subalgebra of Noether symmetries
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ABSTRACT.  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 
$rak{sl}(2,\mathbb{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 $rak{sl}(2,\mathbb{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 
$rak{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 procedure to other types of 
Lagrangians is illustrated.