Cyclic variables for Hamiltonian

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Discussion Overview

The discussion revolves around the Hamiltonian of a single particle in a two-dimensional harmonic oscillator system. Participants explore the concept of cyclic coordinates and constants of motion, questioning the implications of the Hamiltonian's dependence on certain coordinates and the potential for alternative coordinate systems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a Hamiltonian and questions whether the explicit appearance of all coordinates implies the absence of cyclic coordinates and constants of motion.
  • Another participant emphasizes that the Hamiltonian should not be expressed in terms of derivatives of momenta and challenges the interpretation of cyclic coordinates based on the given expression.
  • Further inquiries are made about the existence of other coordinates that could potentially be cyclic, suggesting that there may be alternative representations.
  • Participants discuss the symmetries of the Hamiltonian and the possibility of transforming to polar coordinates, with one participant affirming the system's identification as a two-dimensional harmonic oscillator.
  • A later reply prompts consideration of the consequences of using polar coordinates for the Hamiltonian.

Areas of Agreement / Disagreement

Participants generally agree on the identification of the system as a two-dimensional harmonic oscillator, but there is no consensus on the implications regarding cyclic coordinates and constants of motion. The discussion remains unresolved regarding the potential for alternative coordinate systems.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the nature of cyclic coordinates and the specific properties of the Hamiltonian in different coordinate systems. The exploration of polar coordinates is not fully resolved.

digogalvao
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A single particle Hamitonian ##H=\frac{m\dot{x}^{2}}{2}+\frac{m\dot{y}^{2}}{2}+\frac{x^{2}+y^{2}}{2}## can be expressed as: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{x^{2}+y^{2}}{2}## or even: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{\dot{p_{x}}^{2}+\dot{p_{x}}^{2}}{4}##. Does it mean that the system has no cyclic coordinates? Since all relevant coordinates appear explicitly in ##H##. In that case, there are no constants of motion?
 
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First of all, the Hamiltonian is a function of the generalised coordinates and the corresponding canonical momenta (or more generally, of coordinates on phase space). You should not write the Hamiltonian as a function of derivatives of the momenta. Second, it is unclear to me what "it" that is supposed to mean that the system has no cyclic coordinates is. What property of the expression are you referring to? Clearly, none of the coordinates that you have are cyclic, but that does not mean that there are necessarily other coordinates you could use where one of them would be cyclic (as in fact there is in this case).
 
Orodruin said:
First of all, the Hamiltonian is a function of the generalised coordinates and the corresponding canonical momenta (or more generally, of coordinates on phase space). You should not write the Hamiltonian as a function of derivatives of the momenta. Second, it is unclear to me what "it" that is supposed to mean that the system has no cyclic coordinates is. What property of the expression are you referring to? Clearly, none of the coordinates that you have are cyclic, but that does not mean that there are necessarily other coordinates you could use where one of them would be cyclic (as in fact there is in this case).
What other coordinate is that?
 
digogalvao said:
What other coordinate is that?
What symmetries does the Hamiltonian have? In what coordinates would the Hamiltonian therefore not depend on one of the coordinates?
Hint: What you have is quite clearly a 2-dimensional harmonic oscillator.
 
Orodruin said:
What symmetries does the Hamiltonian have? In what coordinates would the Hamiltonian therefore not depend on one of the coordinates?
Hint: What you have is quite clearly a 2-dimensional harmonic oscillator.
Yes, the frequency is ##\frac{1}{\sqrt{m}}##. Should I put everything in polar coordinates?
 
What happens if you do?
 

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