Can I find a explanation of the Lagrangian formalism for the non-relativistic mechanical theory of particles? I have seen a formalism of the Lagrangian based concepts like mapping variations and I would like to learn it.
I don't understand exactly this procediment:
In general, for an arbitrary function $\Phi$ which depends of time:
$\Phi'(t') = \Phi(t) + \delta\Phi(t)$  (This is exactly what I don't understand)
where $t' = t + \delta t$
It is defined a mapping variation: $\bar\delta \Phi(t)= \Phi'(t)-\Phi(t)$
So $\delta\Phi(t)=\Phi'(t') - \Phi(t) = [\Phi'(t') -\Phi(t')]+ [\Phi(t')- \Phi(t)]$
Using the Taylor's serie in the second part:
$\delta\Phi(t)= \bar\delta \Phi(t') + \frac{d\Phi(t)}{dt}\delta t$
So: $\Phi'(t') = \Phi(t) + \bar\delta \Phi(t') + \frac{d\Phi(t)}{dt}\delta t$
For a $L=L(t): L(t')=L(t)+\bar\delta L(t') + \frac{dL(t)}{dt}.\delta t$
It is demonstrated that: $\bar\delta L(t) = \frac{\partial L}{\partial q_i}\bar\delta q_i(t)+\frac{\delta L}{\delta q'_i}\bar\delta q'_i(t)$
$\bar\delta L(t) $$  =L'(t)-L(t)=L[q'_i(t),\dot q'_i(t),t]-L[q_i(t),\dot q_i(t),t]\\=L[q_i(t)+\bar\delta q_i(t),\dot q_i(t)+\bar\delta \dot q_i(t),t]-L[q_i(t),\dot q_i(t),t]\\$
 A: There are many reference textbooks. I was personally taught by Thornton and Marion's Classical Dynamics during a summer course at Cornell, but several are available from reputable free sources online, too.


*

*Shapiro's preliminary version of Classical Mechanics (PDF),

*Baez's lecture notes on classical mechanics (PDF),

*it's also the starting point for The Structure and Interpretation of Classical Mechanics (HTML), and

*Stone and Goldbart's Mathematics for Physics (PDF).


The basic idea is, consider paths which map the real time interval $(0, T)$ to some coordinates of particles $X = \{x_i(t), i\in \{1, 2,\dots n\}\}$ over this time. Paths form a vector space by the usual approach: you can add them by adding their coordinates independently, you can multiply them by scalars by multiplying all of their coordinates by the scalars independently.
Very often, all of the physics of a system can be expressed in terms of a function $S[X] \in \mathbb R$ by just saying that, of all of the paths that a system could possibly take between two fixed endpoints $X(0)$ and $X(T)$, the ones that it actually does take are stationary inputs for $S$.  In other words, $S[X + \epsilon Y] \approx S[X] + \epsilon^2~T[X, Y, \epsilon]$ for some $T$, the key is that the term proportional to $\epsilon$ vanishes for small $\epsilon$ -- this defines a physically-acceptable path that the system can follow -- the exact details of $T$ are not particularly important for this.
Usually the action principle takes a form like, $$S[X] = \int_0^Tdt~L(x_1(t), x_2(t), \dots x_n(t), \dot x_1(t), \dot x_2(t), \dots \dot x_n(t), t).$$
This function $L$ is called the "Lagrangian", it is a simple function $\mathbb R^{2n+1}\to\mathbb R$ with the usual partial derivatives. Call its first parameters $q_i$ and its next parameters $v_i$ if you wish, to keep one's mind straight about the difference before vs. after substituting in X and Y. After computing $S[X + \epsilon Y]$ we find that the terms linear in $\epsilon$ take the form $$0 = \epsilon~\int_0^T dt~\left[\sum_i\left({\partial L\over \partial q_i}\big(X(t), \dot X(t), t\big)\cdot y_i(t) + {\partial L\over \partial v_i}\big(X(t), \dot X(t), t\big)\cdot y_i(t)\right)\right].$$
After integrating the right-hand terms by parts one can factor out a plain $y_i(t)$, and the vanishing of $y_i(0)$ and $y_i(T)$ means that this does not introduce boundary terms. Since this needs to hold for arbitrary $Y$ we must be multiplying it by a zero-function, and so for each $i$ independently, this results in the "Euler-Lagrange" equations, $${\partial L \over \partial q_i}\big(X(t), \dot X(t), t\big) = \frac{d}{dt}\left[{\partial L \over \partial v_i}\big(X(t), \dot X(t), t\big)\right].$$
