Abstract
Lie's theory of differential equations is applied to the equation of motion of the classical one-dimensional harmonic oscillator. The equation is found to be invariant under a global Lie group of point transformations that is shown to be SL(3, R). The physical significance of the analysis and the results is considered. It is shown that the periodicity of the motion is a local topological property of the equation, while the length of the periods depends upon global properties.