- #1

gionole

- 281

- 24

- Homework Statement
- Show Taylor series for the lagrangian is the following

- Relevant Equations
- ..

We have ##L(v^2 + 2v\epsilon + \epsilon^2)##. Then, the book proceeds to mention that we need to expand this in powers of ##\epsilon## and then neglect the terms above first order, we obtain:

##L(v^2) + \frac{\partial L}{\partial v^2}2v\epsilon## (This is what I don't get).

We know taylor is given by: ##p(x) = f(a) + f'(a)(x-a) + ....##

I tried doing this in respect to $\epsilon$. We know ##\epsilon## is super small, so our taylor is ##f(0) + f'(0)(x-0) = L(v^2) + \frac{\partial L}{\partial \epsilon}(2v+2\epsilon)|(\epsilon=0)(\epsilon-0) = L(v^2) + \frac{\partial L}{\partial \epsilon}2v\epsilon## (note that I don't have ##\frac{\partial L}{\partial v^2}##)

If I try doing taylor with respect to ##v^2##, then I don't get ##L(v^2)## as the first part because ##f(a) != L(v^2)##

Where am I making a mistake?

##L(v^2) + \frac{\partial L}{\partial v^2}2v\epsilon## (This is what I don't get).

We know taylor is given by: ##p(x) = f(a) + f'(a)(x-a) + ....##

I tried doing this in respect to $\epsilon$. We know ##\epsilon## is super small, so our taylor is ##f(0) + f'(0)(x-0) = L(v^2) + \frac{\partial L}{\partial \epsilon}(2v+2\epsilon)|(\epsilon=0)(\epsilon-0) = L(v^2) + \frac{\partial L}{\partial \epsilon}2v\epsilon## (note that I don't have ##\frac{\partial L}{\partial v^2}##)

If I try doing taylor with respect to ##v^2##, then I don't get ##L(v^2)## as the first part because ##f(a) != L(v^2)##

Where am I making a mistake?