# What are examples of classical physical systems having polynomial observables of degree greater than 2?

Specifically: What are empirically well-understood examples of (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\{q^i,p_i\mid i=1,\ldots,n\}$, having degree greater than 2?

Below are follow up questions/replies in response to the comments/questions of Ron Maimon:

With respect to integrable motion in one dimension, what are physical examples of one-dimensional potentials containing polynomial expressions of degree greater than 2?

Beyond the integrable motion of a single particle in one dimension, what are empirically well-understood examples of many-body one-dimensional (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\{q^i,p_i\mid i=1,\ldots,n\}$, having degree greater than 2?

What are well-understood examples of single particle (or many-body) greater than one-dimensional (integrable) Hamiltonian systems whose Hamiltonians include polynomial expressions, in the canonical coordinates $\{q^i,p_i\mid i=1,\ldots,n\}$, having degree greater than 2?

As to the naturalness of these questions, the restriction is to the at least cubic polynomials in the Poisson algebra of classical polynomial observables in the $q^i$ and $p_i$ on phase space (presumably ${\mathbb R}^{2n}$ with $n>1$ in the single particle case). That said, could you expand on your observation concerning the naturalness of this restriction in the context of QFTs?

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Every one dimensional motion is integrable, so any one dimensional potential is an example. This question is strange--- why are you restricting yourself to polynomials? There is only a natural reason to do that in quantum field theory. – Ron Maimon Dec 21 '11 at 3:36