Lagrangian on a saddle advice?

[teeeeee]

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 don't have to worry about moment of inertia.

Does anyone have any advice on where to begin?

Any help will be much appreciated.

Thanks

 

[Ben Niehoff]

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

$$\lambda(t) (z - \tilde x^2 + \tilde y^2)$$

where

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