# Generalized Coordinates and Porn

• A
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

vanhees71
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