# Contact interaction in Lagrangians

Hi all,

If I take an action involving two point particles coupled together by a delta function contact interaction is it possible to carry out the variation with respect to the fields? For e.g.

$S = \int dt \frac{1}{2} \dot{x}^{2} + \int \int dt \dot{x}(t) \delta^{D}\left(x(t) - y(t')\right) \dot{y}(t')dt' + \int dt' \frac{1}{2}\dot{y}^{2}$

working in, say, D dimensions. I think it will be possible to integrate the delta function out in D=1, but not in higher target space dim. The problem I have is in how to do the variation of the delta function term. Physically it is producing an interaction every time the worldlines of the particles intersect and I've tried writing this as a sum over such points - where $x(t_{0})=y(t')$ - of $\frac{\delta(t - t_{0})}{\dot{x}(t_{0})}$ but this is valid only in D = 1 and I still can't get the variation correct.

Any help would be appreciated.
I

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