  Searching for a Lagrangian may seem either a trivial endeavour or an
impossible task. In this paper we show that the Jacobi last multiplier
associated with the Lie symmetries admitted by simple models of classical
mechanics produces (too?) many Lagrangians in a simple way. We exemplify the
method by such a classic as the simple harmonic oscillator, the harmonic
oscillator in disguise [H Goldstein, {\it Classical Mechanics}, 2nd edition
(Addison-Wesley, Reading, 1980)] and the damped harmonic oscillator. This is
the first paper in a series dedicated to this subject.
