The one-dimensional harmonic oscillator has the Lagrangian . Suppose you did not know the...

The one-dimensional harmonic oscillator has the Lagrangian . Suppose you did not know the solution to the motion, but realized that the motion must be periodic and therefore could be described by a Fourier series of the form

(taking t = 0 at a turning point) where ω is the (unknown) angular frequency of the motion. This representation for x(t) defines a many-parameter path for the system point in configuration space. Consider the action integral I for two points, t1 and t2 separated by the period T = 2π/ω. Show that with this form for the system path, I is an extremum for nonvanishing x only if aj = 0, for j ≠ 1, and only if ω2 = k/m.