The Axes not Analyzed, by Jonnie Pober

[This poem is about a pair of pendula, coupled together by a spring. Analyzing this problem by considering the motion the two pendula is very difficult. However, if instead we use a rotated set of axes in Hilbert space, axes corresponding to the motion of the two normal modes (the pendulum mode with eigenvector P, and the breathing mode with eigenvector B, the problem becomes much easier.]

Two masses oscillated on a spring of wood,
And sorry that I could not separate both
And be in Des Cartes’ plane, long I stood
And looked down Hilbert space as far as I could
To where initial positions were plotted on axes’ spokes;
Then looked again, new axes just as fair,
And having perhaps the better claim,
Because normal mode eigenvectors were plotted there;
And as for that the analysis where
Normal modes superpose shall be the game,
Though both axes equally lay
In space no intro-physics work had penciled black.
Oh, I kept the first axes for another day!
Yet knowing inner products of eigenvectors and position lead the way,
I doubted I should need to come back.
I will multiply these coefficients with eigenvectors P and B
And let them oscillate in time for ages and ages hence.
Two masses oscillated on a spring, and me –
I superposed their normal modes with glee
And that has made all the difference.

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