- #1
WWCY
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Homework Statement
I'm having some issues understanding a number of concepts in this section here. I attached the corresponding figure at the end of the post for reference.
Issue 1)
1st of all, I understand that a Hamiltonian can be written as such
$$H = T_2 - T_0 + U$$
whereby ##T_2## and ##T_0## are Kinetic Energy terms quadratically dependent and linearly independent on generalised velocities respectively.
However, the text above writes that ##T_2 = \frac{ma^2}{2 \dot{\theta} ^2}## and ##T_0 = \frac{1}{2}m\omega ^2 a^2 \sin ^2 \theta##.
Since we defined ##r,\theta , \phi## as our spherical coordinates, and ##\omega = \dot{\phi}##, why was the term identified as ##T_0## not considered to be quadratic in velocity dependence?
What constitutes a generalised coordinate/velocity, and what constitutes "something else"?
Issue 2)
So far, I have learned that the azimuthal angle (##\theta## in this case) is to be defined from the "North" end of the ##z## axis. In this case, ##\theta## is defined from the "South" end. Is there any appreciable difference in both definitions?
Assistance is greatly appreciated!