I'm taking a course on mechanical vibrational analysis and this is what I was told by my professor, but I'm curious if there are any counter-examples.
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$\begingroup$ Concerning oscillators, see also physics.stackexchange.com/q/159021/2451, physics.stackexchange.com/q/178375/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Mar 12, 2016 at 17:41
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Consider a particle moving in an $|r|^2$ potential. One possibility is circular motion, which can be viewed as harmonic "vibrations" in orthogonal directions, $\pi /2$ out of phase. Now consider a particle moving in an $|r|^{-1}$ potential. Circular motion is also possible, but is it a "vibration"?
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$\begingroup$ There's no exchange of energy in circular motion. I'm curious about vibrations that involve energy exchange, and if there is any type of vibration where energy is exchanged in other forms than kinetic <-> potential. $\endgroup$ Commented Mar 12, 2016 at 23:44
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$\begingroup$ In electromagnetic waves or LC circuits, energy is exchanged between electric and magnetic fields. This might not meet your definition of "mechanical". $\endgroup$ Commented Mar 13, 2016 at 5:12
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$\begingroup$ Is that an exchange between two different forms of potential energy? $\endgroup$ Commented Mar 13, 2016 at 21:55
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$\begingroup$ A different way of looking at it is that current in an inductive circuit is analogous to momentum. Sorry, I don't think there is a clear cut answer for this. It depends on how you define your terms and your interpretation. $\endgroup$ Commented Mar 14, 2016 at 3:11
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$\begingroup$ If we use electromagnetic radiation as an example, is the exchange of electric and magnetic energy an exchange of two forms of potential energy? $\endgroup$ Commented Mar 15, 2016 at 15:25