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.