How to Perform Classical Mechanic Transformations?

[Mateusz]

Hello everyone, I have a little problem with some transformation.
I wonder how i can get that result. Can somebody explain it step by step?
The " ' " means derivative. Thank you for your time ;)

 

[Geofleur]

Try starting with a really simple case, like if you have just $x_1 = f_1(q_1,q_2)$ and $x_2 =f_2(q_1,q_2)$. You can work out each term in the resulting Lagrangian in this case and that may help you see how it works in general.

 

[loops496]

As Geofleur said start with the simplest but non trivial case, then step it up a little bit, you'll see the pattern right away and convince yourself of the general formula.

 

[Mateusz]