Lagrangian Mechanics: Solving Eqtn 2.28, 2.36, and 2.37

[athrun200]

I don't know how to do the 4 questions.
And I only have some ideas on questions 1.
The details are written on the photos.

Thanks for help.

Eqtn 2.28 2.36 2.37 are given on the photo.

 

[sgd37]

you need to realize that $$F_i = \vec{F}$$ then you can derive your relations

either that or what is the meaning of the i index