Which Textbook References Hamiltonian Systems with Equal Orbit Periods?

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Discussion Overview

The discussion revolves around identifying textbooks that reference Hamiltonian systems with equal orbit periods. Participants explore the implications of closed orbits in Hamiltonian mechanics and seek specific references for their proofs or concepts related to this topic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant states that if a Hamiltonian system has closed orbits, then the periods of all orbits at the same energy level are equal, seeking a textbook reference for this fact.
  • Another participant suggests that this result may be a consequence of the Hamilton-Jacobi partial differential equation, implying that any textbook covering this equation could suffice as a reference.
  • A different participant expresses uncertainty about the relevance of the Hamilton-Jacobi equation to their proof, indicating a desire for a reference without needing to rewrite established concepts.
  • One participant questions the need for a reference if the proof is already known, highlighting a potential misunderstanding of the request.
  • Another participant proposes that a textbook discussing action-angle variables might provide the necessary context, assuming the proof relates to closed orbits.
  • A participant recalls that the theorem in question might be part of an exercise in "Foundations of Mechanics" by Abraham and Marsden, specifically referencing problem 5.2G.
  • One participant confirms that the "period-energy" relation is indeed found on page 198 of the mentioned textbook, aligning with the original request for a reference.

Areas of Agreement / Disagreement

There is no clear consensus on the best reference, as participants propose different sources and approaches. Some express uncertainty about the relevance of the Hamilton-Jacobi equation, while others suggest textbooks on action-angle variables. The discussion remains unresolved regarding the most appropriate reference.

Contextual Notes

Participants express varying levels of familiarity with the Hamilton-Jacobi equation and action-angle variables, indicating potential gaps in assumptions or definitions. The discussion reflects differing interpretations of the relationship between closed orbits and the referenced textbooks.

wrobel
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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
 
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Isn't this a direct consequence of the Hamilton Jacobi partial differential equation? Then one can quote any textbook treating the HJPDE.
 
I need to think my proof does not refer the Hamilton Jacobi partial differential equation
 
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 ;-).
 
vanhees71 said:
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
 
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.
 
wrobel said:
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.
 
Haborix said:
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.
 
Last edited:
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.
 
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Great! this is called "period-energy" relation. p. 198 That is exactly what I asked for. thanks a lot
 
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  • #11
Haborix said:
It is problem 5.2G
solution to the problem:
 

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