Generalized Coordinates and Porn

[observer1]

Yes, that is a serious title for the thread.

Could someone please define GENERALIZED COORDINATES?

In other words (and with a thread title like that, I damn well better be sure there are other words )

1. I understand variational methods, Lagrange, Hamilton, (and all that).
2. I understand the pendulum and the distinction between x/y and r/theta
3. I understand how generalized velocities can depend on generalized coordinates and so on.
4. I understand how they represent the minimum variables needed to describe a system...

OK. But could someone provide a clear, concise definition of the word "generalized?" What makes x/y Cartesian and r/theta "generalized?" When does one have the right to attach the modifier "generalized" to a coordinate system describing a mechanical (or otherwise) system?

What is a generalized coordinate?
(I know it when I see it -- like porn -- but I can't define it.)

 

[wrobel]

Generalized coordinates are local coordinates on configuration manifold

 

[observer1]

Wow... that was good... thanks!

May I ask for one more thing?

It turns out the in classical mechanics, the kinetic energy is not just a function of the generalized velocities. It is also a function of the generalized coordinate.

(As you must well know, KE = 0.5 * m * v * v. But when the coordinates are generalized, the coordinate also appears in the KE.)

In the context of your previous answer, could you demonstrate why this happens?

 

[wrobel]

the kinetic energy is Riemann metric on configuration manifold (precisely speaking, quadric part of the kinetic energy)

Assume we have a particle of mass $m$ moving on a plane. The kinetic energy is $T=m(\dot x^2+\dot y^2)/2$; now express the kinetic energy in terms of polar coordinates: $x=r\cos\phi,\quad y=r\sin\phi$;

 

[observer1]

And, thank you once again!