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

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, 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?