Generalized Coordinates and Porn

  • #1
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Main Question or Discussion Point

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.)
 

Answers and Replies

  • #2
Generalized coordinates are local coordinates on configuration manifold
 
  • #3
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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?
 
  • #4
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##;
 
  • #5
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And, thank you once again!
 

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