I'm a 12th Grader and I'm interested in Lagrangian Mechanics and having a bit of knowledge about the Newtonian Mechanics. So, I found a book of Landau and Lifshitz's Mechanics and started reading from the very first chapter, but I've encountered some serious doubts here!
I'm just writing the symbol/variable meaning and conventions here first!
For instance, let's imagine a particle. $q$ is it's radius vector magnitude (it's scalar), $\dot q$ is the derivative of position vector or velocity (scalar), $t$ as time duration, $S$ as action.
So, $$S = \int \limits_{t_1}^{t_2} L(q, \dot q, t) dt.\tag{2.1}$$
Let variation of function be $\delta q(t)$, so now
$$\delta q(t_1) = \delta q (t_2) =0 \tag{2.3}$$
$$\Rightarrow \qquad\delta S = \int \limits_{t_1}^{t_2} L(q+ \delta q, \dot q + \delta \dot q, t)dt - \int \limits_{t_1}^{t_2} L (q, \dot q, t)dt = 0.\tag{2.3b}$$
So, after the next few lines, it changes to:
$$\delta S = \delta \int \limits_{t_1}^{t_2} L(q, \dot q, t)dt =0.\tag{2.4}$$
This is doubtful as $\delta$ isn't a number which can be multiplied both sides of equal sign both ways,
$$\int \left ( \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot q} \delta \dot q \right)dt = 0.\tag{2.4b}$$
From which multiverse the above thing in concluded even I can't understand after hours of thinking, please help me with these concepts.
Ref: https://archive.org/details/Mechanics_541/page/n11
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