Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on the Lagrangian of a free particle. At the very top of page 7, he equates two Lagrangian by
L′=L(v′2)=L(v2+2v⋅ϵ+ϵ2).
Then he goes to to say that you can expand this in powers of ϵ and neglect all terms above the first order to get
L(v′2)=L(v2)+∂L∂v22v⋅ϵ.
How do you expand in terms of ϵ to get this result? This Lagrangian seems to me to be a function of three variables and I am looking at the multivariable Taylor expansion and I am not seeing how Landau could get his result.
Answer
Landau defines v′≡v+ϵ. Then
L(v′2)=L[(v+ϵ)2]=L(v2+2v⋅ϵ+ϵ2)∼L(v2+2v⋅ϵ)
In the last step we have omited the second order terms in ϵ.
So we have L(v2+2v⋅ϵ) which is a function like f(a+x). We expand it as
f(a+x)∼f(a)+f′(a)x
at first order. In our case, we have a=v2 and x=2v⋅ϵ, so
L(v′2)∼L(v2+2v⋅ϵ)∼L(v2)+∂L∂v22v⋅ϵ
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