`In  this  question a  nontrivial   periodic  orbit  is  a  periodic  orbit  which  is not a  singular point.`

Let  $p: \mathbb{R}^n \to  \mathbb{R}$  be  a  polynomial function.  We  define  the    Hamiltonian $H$  on  $\mathbb{R}^n \times \mathbb{R}^n$ as  follows:

$$H(x,y)= \sum_{\alpha}\frac{1}{\alpha!} D^{\alpha} p(x) y^{\alpha}$$  where  $\alpha$  varies  over  all  multi integer indices  and  $x,y \in \mathbb{R}^n.$

What  can  be  said about the  dynamics  of  the  corresponding  Hamiltonian  vector  field? 

 **Is there  an  example  of  such a  Hamiltonian  vector  field which  possess a  non trivial periodic  orbit?**

We  observe that  the  function $\sum x_i + \sum y_i$ is  a  first integral.  Are there  some  other first integrals, independent of $\sum x_i + \sum y_i$? In particular is this Hamiltonian completely integrable?


Now  we  try to  extend this questions on an arbitrary  Riemannian  manifold. So  we consider a kind of Taylor series and we cut the  series  at its  second  term.  So our  question would  be  the  following:

>Let  $(M,g)$  be  a  Riemannian  manifold  and  $f:M \to  \mathbb{R}$ be  an  smooth  map.  We  define the  following Hamiltonian on the  tangent  bundle  $TM$  with its  obvious  symplectic  structure

>$$H(x,v)=f(x)+df(x).v +\frac{1}{2} Hess(f)(x)(v,v)$$
where $Hess(f)$ is  the  $2-$ linear form on $T_x M$  with the  formula $$Hess(f)(x)(v,w)=g(\nabla_v \nabla f, w)$$  where $\nabla$  is  the  LC  connection associated  with the  Riemannian  metric.
>Is  there an  example of such  a  Hamiltonian with a non trivial  periodic  orbit?