# Hamiltonian mechanics: phase diagram

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.

Last edited:

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

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
Gold Member
You can get it by transformation from Descartes coordinates.

Dr Transport
Gold Member
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.

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
Gold Member
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