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Does there exist a polynomial Hamiltonian function $H$ on some $\mathbb{R}^{2n}$ such that

  1. Any polynomial function $P$ such that $\{P,H\}=0$ is of the form $p(H)$ for some polynomial $p$ in one variable;
  2. There exists a smooth function $F$ such that $\{F,H\}=0$, and yet $F$ is not of the form $f(H)$ for any smooth function $f$ in one variable?

I am looking for an example of such a phenomenon, or a proof that it cannot occur. A non-Hamiltonian example -- that is, of a polynomial vector field such that the smooth completion of the algebra of its polynomial integrals is strictly smaller than the algebra of its smooth integrals -- would also be helpful. Any references to the literature will be much appreciated, of course.

Thanks in advance!

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For the non-Hamiltonian case, take the ODE $\dot x = (x^2+1)$ and $\dot y = 2xy$ in the plane. This is a polynomial vector field on the plane, and it has a first integral $y/(x^2+1)$, which is not a polynomial. Moreover, it is easy to see that there is no polynomial first integral $P(x,y)$ of this ODE that is not constant. – Robert Bryant Jul 12 '11 at 12:14
Dear Robert, your comment is very helpful and counts as an answer. Thanks! – Dmitry Roytenberg Jul 13 '11 at 8:47
up vote 2 down vote accepted

Here is a tentative Hamiltonian answer having 2 degrees of freedom. Take $H = (1/2)(1 + a x^2 + bxy + cy^2) (p_x ^2 + p_y ^2)$. for essentially any parameters $a, b, c$ for which
$ax^2 + bxy + cy^2$ is positive definite ($a x^2 + bxy + c y^2) > \epsilon (x^2 + y^2)$) but NOT a multiple of $x^2 + y^2$. This $H$ is the Hamiltonian for geodesic flow on the plane with metric $ds^2 = (dx^2 + dy^2)/((1 + a x^2 + bxy + cy^2)$. Comparing the arclength $ds$ with $dr/\sqrt{1 + \epsilon r^2}$ seems to show that the metric is complete. Now we can play the scattering integrability' trick which I learned from E. Gutkin. This trick yields 2 new integrals $F, G$ as thescattering data'' for the resulting geodesics. What I mean is that any geodesic will be asymptotic to a straight line in the xy plane, and the asymptotic direction $(F, G)$ of this line is an integral. The hard part is to show this asymptotic direction, is a smooth function of $x, y, p_x, p_y$. I wager that unless $a x^2 + bxy + cy^2$ is a square (like $x^2$) or a multiple of $x^2 + y^2$, these scattering integrals are not polynomials. They might not even be analytic..

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There are such examples (with transcendental integrals of motion) already on $\mathbb{R}^{4}$, see the paper of Hietarinta.

The Hamiltonian is $$H=p_x^2/2+p_y^2/2+2y p_x p_y-x,$$ the desired integral is $$I_1=p_y \exp(p_x^2).$$

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