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

Question

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?

Answer 1

A way to generate integrable polynomial Hamiltonian systems of any desired degree is to take your favorite nonintegrable system and expand it in a normal form about some known solution. In general the procedure requires several canonical transformations in order to go from say Cartesian coordinates to normal coordinates and in general diverges. But, one may truncate the expansion at some predetermined polynomial degree. The result is an integrable polynomial Hamiltonian system. If desired you can invert the canonical transformations to go back to your original coordinates. I learned the technique from this: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (Applied Mathematical Sciences) Meyer,Hall. Hope this helps.