I am confused from the equation 6, why we get Euler-Lagrange equation from equation 8 but not from equation 6?
Why we need to use $\zeta$ as invariant parameter in equation 8 even we already have invariant parameter $s$ in equation 6, in order to get relativistic euler-lagrange equation?
Reference: Relativistic mechanics satya prakash page no.402
Assuming that we are talking about a massive point particle, we know that the arclength $$s~=~c\tau\tag{A}$$ is the speed of light $c$ times the proper time $\tau$ (up to an additive constant), and the 4-velocity $$u^{\mu}~:=~\frac{dx^{\mu}}{d\tau}\tag{B}$$ satisfies $$u^{\mu}u_{\mu}~\stackrel{(A)+(B)+(3)}{=}~c^2\tag{7}.$$ [For the overall sign, compare with the Minkowski sign convention (3).]
The most important point (which Prakash doesn't seems to explain) is now that in the stationary action/Hamilton's principle (in contrast to e.g. Maupertuis' principle) the integration region $[\zeta_1,\zeta_2]$ for the world-line parameter $\zeta$ is kept fixed and the same for all paths/trajectories.
Also note that the 4 position coordinates $x^{\mu}$ are to varied independently (say, within timelike curves), and that the quantity $$ \dot{x}^{\mu}\dot{x}_{\mu}, \qquad \dot{x}^{\mu}~:=~\frac{dx^{\mu}}{d\zeta}, \tag{C}$$ is not fixed (but say, positive).
The main reason that we cannot pick the arclength $s$ (or equivalently the proper time $\tau$) as the world-line parameter $\zeta$ is that the integration region $[s_1,s_2]$ should then be fixed, but this contradicts the fact that neighboring paths/trajectories clearly generically have different arclengths.
Moreover, Prakash points out that if $\zeta=\tau$ then the 4 position coordinates $x^{\mu}$ cannot be varied independently because of the constraint (7), cf. eqs. (A)+(B)+(3), i.e. there are only 3 independent position variables, so the variational principle (in its current form) does also not work for this reason.
