# Mechanics by Landau

Hi!

Just reading the first book by Landau in the theoretical physics course, and I need some guidance about one point (cf. 4, Lagrangian for a free particle.) Notations: L and L' are Lagrangians refered to different inertial frames of reference. e is a element of velocity between L and L'.
It says " We have L' = L (v'^2) = L (v^2 + 2ve + e^2 ). Expanding this expression in powers of e and neglecting the terms above the first order, we obtain

L (v'^2) = L (v^2) + partial derivative of L respect to v^2 times 2ve​

So, if this is supossed to be a Taylor serie with n from zero to one, why appears the derivate of L with respect to v^2. Can someone, please, make all steps explicit?

Any help would be very appreciated!

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dx
Homework Helper
Gold Member
L is a function of v², and when you change to the new intertial frame v² becomes v² + 2v⋅ε to first order in ε because (v + ε)² = v² + ε² + 2v⋅ε . The change in L to first order in ε is therefore (∂L/∂v²)(2v⋅ε).

Thanks for the post.
I don't see why is not (∂L/∂v²)(v²+2v⋅ε)

dx
Homework Helper
Gold Member
If you have a function f and you want the change in f from a to a + Δ, you have to multiply f'(a) by Δ, i.e. f(a + Δ) = f(a) + f'(a)Δ.

In our case, v² becomes v² + 2v⋅ε, so you multiply by 2v⋅ε.

Thanks for your help, clear now! Should I study calculus of variations in order to continue with the book?

dx
Homework Helper
Gold Member
No it's not really necessary. The only place where it is used in this book is in section 2, where the Euler-Lagrange equations are derived, and a quite clear description of it is given there.