- #1

gionole

- 281

- 24

Assume ##y(x)## is a true path and we do perturbation as ##y(x) + \epsilon \eta(x)##. From this, it's clear that we arrive at first order variation, assuming that we don't perturb the ##x## domains, but only ##y##. So below is the general formula.

##\delta J = \int_{x_0}^{x_1} \eta(\frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'}) dx + \frac{\partial F}{\partial y'}\Bigr|_{x_1} \eta(x_1) - \frac{\partial F}{\partial y'}\Bigr|_{x_0} \eta(x_0)##

Assume now that I just take ##\eta(x)## to be ##1##, which means I can replace it above and get:

##\delta J = \int_{x_0}^{x_1} (\frac{\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'}) dx + \frac{\partial F}{\partial y'}\Bigr|_{x_1} - \frac{\partial F}{\partial y'}\Bigr|_{x_0} ## (1)

**Question 1**: is the ##\delta J## in (1) the variation difference in first order between the functional of ##y(x)## and ##y(x) + \epsilon## ? I need to confirm this, so yes or no would be great. If no, why ?

**Question 2:**If the answer to question 1 is "yes", then why am I failing at deriving the same ##\delta J## by applying variational principle directly(without replacing) ? i.e assume ##y(x)## is a true path, if so, varied path ##y(x) + \epsilon##'s action still should have no difference with original action in first order(note ##\epsilon## is infinetisemal), but I end up with ##\int_{x_0}^{x_1} \frac{\partial F}{\partial y} dx## which is clearly different.

I probably have some other questions and finally approaching the final understanding of things(at least, i am trying my best and reading an actual book). Thank you for your understanding.