Hamiltonian mechanics: phase diagram

[LuccaP4]

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.

 

[anuttarasammyak]

As a start how about writing down Hamiltonian in Descartes x,y,z coordinates ?

 

[LuccaP4]

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.

 

[anuttarasammyak]

You can get it by transformation from Descartes coordinates.

 

[Dr Transport]

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 x,y,z. 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.

 

[LuccaP4]

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 x,y,z. 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.

 

[vanhees71]

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 x,y,z. 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.

 

[Dr Transport]

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.