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Have you tried plugging your Lagrangian into the Lagrange equation? You should get equations of motion out.

Are you looking for the equation of motion?

I'll say right now that your lagrangian isn't correct. You're differentiating x and y with respect to alpha. You should be differentiating them with respect to t.



But even if we did this we'd get the wrong Kinetic energy. The problem is with your x and y components. Remember that the block slides, so you've only given their positions at t = 0. You have to redefine L as the distance along the hypotenuse between the mass and the thin end of the wedge. And also remember that the wedge itself slides, which possibly makes the issue of degrees of freedom more complicated.

These are the type of questions my students often have to deal with (I'm tutoring (for a ridiculous hourly rate) at the moment 'til I decide what postgrad I want to undertake). So I'll head for now, but if yer still having problems with this question I'll walk you through it tomorrow.