Hamiltonian mechanics: phase diagram

  • #1
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Homework Statement:
Write the Hamiltonian and Hamilton equations for an isotropic three-dimensional harmonic oscillator in spherical and cylindrical coordinates. Build the phase diagram.
Relevant Equations:
..
The issue here is that I don't know how to operate the final equations in order to get the phase diagram. I suppose some things are held constant so I can get a known curve such as an ellipse.
I attach the solved part, I don't know how to go on.

82b7dc783b13b50dc3ca40487b662bda.png


15149cb5c8aaf6a5e8f17f9e9c406f07.png
 
Last edited:

Answers and Replies

  • #2
anuttarasammyak
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As a start how about writing down Hamiltonian in Descartes x,y,z coordinates ?
 
  • #3
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As a start how about writing down Hamiltonian in Descartes x,y,z coordinates ?
But the statement says to write it in spherical and cylindrical coordinates. I have to build the phase space in that coordinates.
 
  • #4
anuttarasammyak
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You can get it by transformation from Descartes coordinates.
 
  • #5
Dr Transport
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As a start how about writing down Hamiltonian in Descartes x,y,z coordinates ?
But the statement says to write it in spherical and cylindrical coordinates. I have to build the phase space in that coordinates.

This is a fundamental skill in physics that isn't taught very well anymore, think of it this way, you don't think in terms of spherical or cylindrical coordinates, you think in terms of [itex]x,y,z[/itex]. The best way to start for nearly any problem is to write all your quantities in Cartesian coordinates and convert. I used to make my students in electromagnetics do this and if they didn't took major points off.
 
  • #6
24
9
You can get it by transformation from Descartes coordinates.
This is a fundamental skill in physics that isn't taught very well anymore, think of it this way, you don't think in terms of spherical or cylindrical coordinates, you think in terms of [itex]x,y,z[/itex]. The best way to start for nearly any problem is to write all your quantities in Cartesian coordinates and convert. I used to make my students in electromagnetics do this and if they didn't took major points off.
Okay, then I'll do that. Thanks.
 
  • #7
vanhees71
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This is a fundamental skill in physics that isn't taught very well anymore, think of it this way, you don't think in terms of spherical or cylindrical coordinates, you think in terms of [itex]x,y,z[/itex]. The best way to start for nearly any problem is to write all your quantities in Cartesian coordinates and convert. I used to make my students in electromagnetics do this and if they didn't took major points off.
Really? The use of the mos convenient coordinates for a given problem is a skill physicists cannot learn early enough.

What's however important to stress is where the used coordinates have singularities and to be careful with expressions like ##\Delta \vec{A}## for a vector field ##\vec{A}## when calculated not in Cartesian coordinates.
 
  • #8
Dr Transport
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Really? The use of the mos convenient coordinates for a given problem is a skill physicists cannot learn early enough.

I agree, on the other hand an undergraduate hasn't enough experience setting up problems. In highly symmetric cases, sure, but anything else, why take the chance of going off into left field.

I was taught to do it that way from some really fine physicists in my formative years and that is the way I taught it, it works. I'm a mere mortal, I do what works to at least set up the problem correctly. Take for example, the Lagrangians for this problem, I couldn't write them down from memory, but if I started with the Cartesian versions, I'd have them written in less than half a page. Same for electro-static/magneto-static problems.
 

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