Harmonic oscillations are isochronous, but so are oscillations in any potential that is shear-equivalent to a harmonic potential. We demonstrate the converse: that every possible isochronous potential is shear-equivalent to a harmonic potential. To do this we show that every potential is shear-equivalent to a unique symmetric potential, and that every isochronous symmetric potential is harmonic. Shearing is thus the only possible period-preserving transformation for potentials with fixed energy.
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We do not demand a positive concavity: the potential could have a vanishing second derivative at its minimum. While we do not explicitly consider the unlikely case of a vanishing second derivative on a neighborhood of the minimum (a horizontal line segment), the argument can be extended to cover this case with minor modification.
9.
The maxima in between them could be symmetrized as well.
© 2018 American Association of Physics Teachers.
2018
American Association of Physics Teachers
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