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We know from the theory of the quantum harmonic oscillator that the energy spectrum can be determined nearly effortlessly once we are aware of the simple algebraic structure.

In a certain sense, we can regard quantum mechanics as a deformation of the classical algebra. This is why I was wondering: can algebraic methods be just as helpful in classical mechanics as they are in quantum mechanics?

Since I suspect that this might generally not the case (algebraic approaches do not seem to be very popular in classical mechanics), can you point me to the reason what happens in the classical limit which makes the algebraic structure less helpful (or even useless)?

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