Simple Harmonic Motion
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Exercise:
The differential of a simple harmonic oscillator ddot y -omega_^ y can be erpreted as a system of two linear first-order differential s. The eigenvalues and eigenvectors are given by lambda_ pm iomega_ vec v_ pmatrix mp i omega_ pmatrix vspacemm Show that the function yt sinomega_ t is the solution for the initial conditions y quad textrmand quad dot y omega_
Solution:
The solution y_t is a superposition of the fundamental solutions: pmatrix y_t dot y_t pmatrix a_ vec v_ e^lambda_ t + a_ vec v_ e^lambda_ t For t we find y_ a_ -i+a_ i -i a_-a_ Longrightarrow a_ a_ dot y_ omega_ a_ omega_ + a_ omega_ omega_ a_+a_ Longrightarrow a_ + a_ a_ Longrightarrow a_ a_ frac It follows that y_t fracleft-i e^iomega_ t+i e^-iomega_ tright -fracilefte^iomega_ t-e^-iomega_ tright -fraci isinomega_ t -i^ sinomega_ t sinomega_ t
The differential of a simple harmonic oscillator ddot y -omega_^ y can be erpreted as a system of two linear first-order differential s. The eigenvalues and eigenvectors are given by lambda_ pm iomega_ vec v_ pmatrix mp i omega_ pmatrix vspacemm Show that the function yt sinomega_ t is the solution for the initial conditions y quad textrmand quad dot y omega_
Solution:
The solution y_t is a superposition of the fundamental solutions: pmatrix y_t dot y_t pmatrix a_ vec v_ e^lambda_ t + a_ vec v_ e^lambda_ t For t we find y_ a_ -i+a_ i -i a_-a_ Longrightarrow a_ a_ dot y_ omega_ a_ omega_ + a_ omega_ omega_ a_+a_ Longrightarrow a_ + a_ a_ Longrightarrow a_ a_ frac It follows that y_t fracleft-i e^iomega_ t+i e^-iomega_ tright -fracilefte^iomega_ t-e^-iomega_ tright -fraci isinomega_ t -i^ sinomega_ t sinomega_ t