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A Generalized Coordinates and Porn

  1. Jun 30, 2016 #1
    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.)
     
  2. jcsd
  3. Jun 30, 2016 #2
    Generalized coordinates are local coordinates on configuration manifold
     
  4. Jun 30, 2016 #3
    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?
     
  5. Jun 30, 2016 #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##;
     
  6. Jun 30, 2016 #5
    And, thank you once again!
     
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