The usefulness of the action-angle variables

[wdlang]

i am now reading the book by goldstein on classical mechanics

classical mechanics is always a difficult subject for me

as for the action angle variables, i have some questions.

If a system is chaotic, so that a trajectory may never close

in this case, how can we define an action variable. In his book, the actioni s defined along a closed orbit

actually, for an arbitrary central field, the orbit may never close, then in this case, the action can not be defined!

therefore, it seems that the action angle variables can only be defined for some particular system.

 

[Valerie Park]

In fact, for chaotic systems, the action angle variables have been used to study the dynamical behavior. For example, one can define an averaged action variable by taking an ensemble average of the action over the chaotic trajectory. This can give us some insight into the long-term dynamical behavior of the system.

 

[charng]

I understand your questions and concerns about the usefulness of action-angle variables. While it is true that these variables may not be applicable to all systems, they have proven to be extremely useful in understanding and describing the behavior of many physical systems. In particular, they are most effective in describing systems that exhibit periodic or nearly periodic motion, such as planetary orbits or simple harmonic oscillators.

In cases where a trajectory may never close, it is still possible to define an action variable by considering the average motion of the system over time. This can provide valuable insights into the dynamics of the system, even if it is chaotic. Furthermore, the use of action-angle variables allows for a more elegant and concise description of the system's behavior, making it easier to analyze and understand.

It is true that action-angle variables may not be applicable to all systems, but this does not diminish their usefulness in the systems where they can be applied. As with any scientific concept, it is important to understand its limitations and when it is most appropriate to use. In the case of classical mechanics, action-angle variables are a powerful tool for understanding and describing the behavior of physical systems and should not be discounted based on their limited applicability. I would encourage you to continue reading and exploring this topic in Goldstein's book, as it is a fundamental concept in classical mechanics that has proven to be incredibly useful in many areas of physics.