**Disclaimer**: I don't know if this question is well suited for this site, but I have posted this question on [Math.StackExchange][1] with no answer, so I have thought to post it even here.


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I am reading about the energy-period relation for Hamiltonian Systems.  
In Weinstein's formulation (cf. Abraham, Marsden, Foundations of Mechanics 2nd Ed, page 198) this relation amounts to:
>$(\ast)$ Given an Hamiltonian system $(M,\omega, H)$, let be $\Phi$ the flow of $X_H$ and $\text{per}_H:=\{(t,x)\mid\Phi(t,x)=x\}.$  
If $N$ is a smooth submanifold contained in $\text{per}_H,$ then $\left.dt\wedge dH\right|_N=0,$ i.e. $t=t(H)$ on $N,$ (the period depends only on the energy.)

In Guillemin, Stenberg, Geometric Asymptotics, on page [170][2], I have additionally found that, when all integral curves of $X_H$ are closed, we can take $N=\text{per}_H$ in $(\ast),$ which should mean that in such a case $\text{per}_H$ is a smooth submanifold of $\mathbb R\times M.$  

Starting from this I was wondering myself:

>If all integral curves of $X_H$ are closed, and $\tau(p)$ denotes the period of the orbit through $p,$ then $\tau:M\to\mathbb R$ is smooth? otherwise how to prove that in such a case $\text{per}_H$ is a submanifold?

My guess is that:  
if there were principal bundle structure $(M,p,X,\mathbb S^1)$ such that the $\mathbb S^1$ orbits are the trajectories of $X_H$ then the period $\tau:M\to\mathbb R$ should be smooth because of the relation $X_H=\tau\zeta,$ where $\zeta$ is the infinitesimal generator of the action.


  [1]: http://math.stackexchange.com/questions/179182/about-the-energy-period-relation-for-hamiltonian-systems
  [2]: http://books.google.it/books?id=58PgdwJzirUC&lpg=PP1&ots=GDIGlP2eaz&dq=guillemin%20sternberg%20Geometric%20Asymptotics&pg=PA170#v=onepage&q&f=false