So what I've done is said if we have a set of coordinates say

[tex] \left{ q_{i} \right\} , i = 1, \ldots ,N[/tex]

where I'll assume this first set is adequate and complete. Since we want to show the tranformation of coordinates and the Lagrangian is invariant, I create another set of the form

[tex] q_{k} = \tilde{f}_{k} (q_{1}^{\ast}, \ldots , q_{N}^{\ast},t) [/tex]

where we assume an invertible relationship defined by above. So the Lagragian becomes

[tex] L ( q_{i},t) = L ( \tilde{f}_{1} (q_{k}) , \ldots , \tilde{f}_{N} (q_{k}) ,t) \equiv L^{\ast} [/tex]

Now the Lagragian equations are of course

[tex] \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_{i}} = \frac{\partial L}{\partial q_{i}} [/tex]

So, I think we must also have

[tex] \frac{d}{dt} \frac{\partial L^{\star}}{\partial \dot{q}_{i}} = \frac{\partial L^{\ast}}{\partial q_{i}^{\ast}} [/tex]

which leads me to my question. I am sure the last equation is correct just by the form, I'm failing to see how to relate the coordinates and the derivatives as they are transformed. Any ideas would be great thanks.