I Which Textbook References Hamiltonian Systems with Equal Orbit Periods?

[wrobel]

There is a nice fact. It approximately sounds like that: Let $H=H(p,q)$ be a Hamiltonian system with $n$ degrees of freedom such that all its orbits are closed. Then the periods of all the orbits belonging to the same energy level are the same.

Please which textbook does contain this? I know the proof I need only reference

 

[vanhees71]

Isn't this a direct consequence of the Hamilton Jacobi partial differential equation? Then one can quote any textbook treating the HJPDE.

 

[wrobel]

I need to think my proof does not refer the Hamilton Jacobi partial differential equation

 

[vanhees71]

Well, if you have an own proof, why do you need a reference? It's hard to give a reference for a proof, you haven't shown to me ;-).

 

[wrobel]

Well, if you have an own proof, why do you need a reference?

because I do not want to enlarge the volume of my text by rewriting of well-known things
And I have no idea how to employ the Hamilton Jacobi eq. which is local in its nature

 

[Haborix]

Would a textbook with a decent treatment of action-angle variables be enough? I assume the proof is along those lines given the reliance on closed orbits.

 

[vanhees71]

because I do not want to enlarge the volume of my text by rewriting of well-known things
And I have no idea how to employ the Hamilton Jacobi eq. which is local in its nature

Then I don't understand what you want to prove.

 

[wrobel]

Would a textbook with a decent treatment of action-angle variables be enough? I assume the proof is along those lines given the reliance on closed orbits.

I think the existence of the action-angle variables needs to be proved first
I just actually asked for a reference I did not have an intention to organize a challenge :)

Thank you everybody.

 

[Haborix]

I couldn't recall anything framed in the way you put it, so I was hoping it might be a corollary of a theorem more common. I think the theorem you describe is part of an exercise in "Foundations of Mechanics" by Abraham and Marsden. It is problem 5.2G on p. 401 in the second edition.

 

[wrobel]

Great! this is called "period-energy" relation. p. 198 That is exactly what I asked for. thanks a lot

 

[wrobel]

It is problem 5.2G

solution to the problem: