The one-dimensional harmonic oscillator has the Lagrangian L = mx˙2 − kx2/2. Suppose you did not know the solution of the motion, but realized that the motion must be periodic and therefore could be described by a Fourier series of the form x(t) =∑j=0 aj cos jωt, (taking t = 0 at a turning point) where ω is the (unknown) angular frequency
The one-dimensional harmonic oscillator has the Lagrangian
L = mx˙2 − kx2/2. Suppose you did not know the solution of the motion, but realized that the motion must be periodic and therefore could be described by a Fourier series of the form
x(t) =∑j=0 aj cos jωt,
(taking t = 0 at a turning point) where ω is the (unknown) angular frequency of the motion. This representation for x(t) defines 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.
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