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Lagrangian on a saddle advice?

  1. Aug 12, 2011 #1
    Hi,

    I am trying to obtain a Lagrangian for a particle moving on the surface of a saddle
    z = x^2 - y^2

    I have an added complication that the saddle is rotating with some angular frequency, w, and not sure how to incorporate this rotation into my kinetic and potential terms.

    This is the kind of thing I am trying to model:
    http://www.fas.harvard.edu/~scidemos/OscillationsWaves/SaddleShape/SaddleShape.html
    But thought best to assume a particle for now, so dont have to worry about moment of inertia.

    Does anyone have any advice on where to begin?

    Any help will be much appreciated.

    Thanks
     
  2. jcsd
  3. Aug 12, 2011 #2

    Ben Niehoff

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    Science Advisor
    Gold Member

    The most straightforward way I can think of for such a system is to write down the ordinary kinetic term in R^3, with zero potential term, and then implementing "the particle stays attached to the saddle" as a constraint (i.e., with a Lagrange multiplier). The constraint will look something like

    [tex]\lambda(t) (z - \tilde x^2 + \tilde y^2)[/tex]

    where

    [tex]\begin{align*} \tilde x &= \cos (\omega t) \; x - \sin (\omega t) \; y \\ \tilde y &= -\sin (\omega t) \; x + \cos (\omega t) \; y \end{align*}[/tex]
     
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