Dynamics of fiberwise starshaped hypersurface of Hamiltonian flows on $T^*M$

Question

I have started reading the following paper arXiv link on Dynamical Systems and Symplectic Geometry and in page $3$ we have the following statement :

Let $\Sigma$ be a fiberwise starshaped Hypersurface in $T^*M$ and $H:T^*M\rightarrow \mathbb{R}$ such that $\sigma=H^{-1}(1)$ and $1$ is a regular value.Then the orbits of $\phi_H^t|_{\Sigma}$ do not depend on the choice of choice of $H$, in the sense that if we pick another $G$ such that $\Sigma=G^{-1}(1)$ and such that $1$ is a regular value, then the flow $\phi_G^t|_{\Sigma}$ is a time change of the flow of $\phi_{H}^t|_{\Sigma}$, i.e , there exists a smooth positive function $\sigma(t,x):\mathbb{R}\times \Sigma\rightarrow \mathbb{R}$ such that $\phi_G^t|_{\Sigma}=\phi_H^{\sigma(t,x)}|_{\Sigma}$.

Does anyone know a reference where I could check why this is true ? I have been thinking about this for a while and can't seem to come up with a proof myself.

Any help is appreciated. Thanks in advance.

Answer 1

Okay after thinking about this again I think I know why this is true. Since $\Sigma$ is Fiberwise starshaped then the Liouville form $\lambda$ of the cotangent bundle will be a contact form on $\Sigma$, and we will have that $d\lambda=\omega$ restricted to $T\Sigma$ will have rank $2n-2$ and it's kernel will be $1$-dimensional. Now by definition of the Hamiltonian vector fields we will have that $0=dH(.)|_{T\Sigma}=\omega(X_H|_{\Sigma},.)$ and the same happends with $X_G|_{\Sigma}$ and so since the kernel in $T\Sigma$ is one-dimensional we have that there exits a smooth function $\sigma(x)$ such that $X_H(x)=\sigma(x)X_G(x)$ for $x\in \Sigma$. And so if $\phi_G^t$ is the Hamiltonian flow of $G$ we have that $\phi_G^{f(t,x)}$ , where $f(t,x)=\int_{0}^{t}\sigma(x)dx$ is the hamiltonian flow of $H$ since $\dot \phi_G^{f(t,x)}=\dot f(t,x)X_G(x)=\sigma(x)X_G(x)=X_H(x)$.