Euler-Lagrange equations in relativity (Goldstein) In order to have a  covariant formulation of special relativity, we stop using the time $t$ as a parameter and we choose some invariant parameter.
In Goldstein (third edition), chapter $7.10$, it goes through this derivation making an argument about why proper time $\tau$ can't be the parameter we are looking for because of the constraint on 4-velocities.
\begin{equation}
u_\nu u^\nu=c^2 \tag{1}
\end{equation}
It then chooses another parameter $\theta$ and derives Euler Lagrange equations using this new parameter $\theta$. At the end he chooses
\begin{equation}\theta=\tau\tag{*}\end{equation}
Why does this allow us to ignore the constraint $(1)$?
We're basically using proper time the whole time but ignoring the constraint and using at the end.
I've read something similar on another book (which used $s=\int ds$ where $ds^2=dx^\nu dx_\nu$, thought) which said that to find the variation of action:
\begin{equation}
S=\int Lds
\end{equation}
We should also consider the variation of $ds$ with coordinates. Then it introduces a new parameter that is eventually replaced by $s$. How do you account for that?
 A: *

*Since the 4-velocity $$u^{\mu}~:=~\frac{dx^{\mu}}{d\tau} \tag{U}$$ by definition is the 4-position differentiated wrt. proper time $\tau$, the condition
$$\begin{align}
c^2\mathrm{d}\tau^2
~=~&\pm \eta_{\mu\nu}\mathrm{d}x^{\mu} \mathrm{d}x^{\nu}
\cr~\Updownarrow~&\cr
 u^{\mu}  u_{\mu}~=~&\pm c^2 \end{align}\tag{C}$$
holds independently of any worldline (WL) parametrization $\theta$ [in Minkowski signature $(\pm,\mp,\mp,\mp)$, respectively].


*Goldstein considers the action for a massive relativistic point particle
$$\begin{align} I ~=~&\int_{\theta_i}^{\theta_f} \! d\theta ~\Lambda, \cr  
\Lambda~=~&-mc\sqrt{\pm x^{\prime}_{\mu}x^{\prime \mu}}, \cr 
x^{\prime \mu}~:=~\frac{dx^{\mu}}{d\theta},\end{align}\tag{I}$$
which is WL reparametrization invariant $\theta\to\tilde{\theta}=f(\theta)$.


*It is important to realize that the 4 quantities $x^{\prime \mu}$ are not constrained by eq. (C). ($\leftarrow$ This is OP's main question.)


*The corresponding Euler-Lagrange (EL) equation$^1$
$$ \frac{d}{d\theta}  \left( \frac{mc x^{\prime}_{\mu}}{\sqrt{\pm x^{\prime}_{\mu}x^{\prime \mu}}}\right)~\approx~0\tag{EL} $$
is WL reparametrization covariant.


*After the variation, it is legitimate to choose the parametrization $\theta=\tau$. The EL equation then simplifies [with the help of eq. (C)] to
$$ \frac{d(m u_{\mu})}{d\tau} ~\approx~0,\tag{EL'}$$
i.e. the 4-acceleration is zero on-shell.


*Goldstein makes the pragmatic observation that if we a priori choose $\theta=\tau$ in the Lagrangian $\Lambda$ [and somehow ignore the constraint (C)], then we can formally write down the correct EL-equation [with $\theta=\tau$].


*Although Goldstein obtains the correct EL-equation by this dirty trick, it is conceptionally very misleading. In fact, if we literally choose the parametrization $\theta=\tau$ in the action prior to the variation then the action would become a constant off-shell:
$$ \begin{align} I ~=~&\int_{\tau_i}^{\tau_f} \! d\tau~(-mc)\sqrt{\pm u_{\mu}u^{\mu}}\cr
~=~&-mc^2(\tau_f-\tau_i).\end{align}\tag{I'}$$
Phrased differently: all virtual paths would have the same value, i.e. the stationary action principle becomes ill-defined.
References:

*

*H. Goldstein, Classical Mechanics, 2nd edition; Section 7.9.


*H. Goldstein, Classical Mechanics, 3rd edition; Section 7.10.
--
$^1$ The $\approx$ symbol means equality modulo EOM.
A: Proper time $\tau$ can be defined as the parametrization of $x^\mu(\tau)$ such that :
$$u_\mu u^\mu = \frac{\text d x_\mu}{\text d\tau}\frac{\text d x^\mu}{\text d\tau} = c^2$$
If you choose an arbitrary parametrization $x^\mu(\theta)$, then you have :
$$\frac{\text d x_\mu}{\text d\theta}\frac{\text d x^\mu}{\text d\theta} = \left(\frac{\text d\tau}{\text d\theta}\right)^2 \frac{\text d x_\mu}{\text d\tau}\frac{\text d x^\mu}{\text d\tau} = \left(\frac{\text d\tau}{\text d\theta}\right)^2  c^2$$
so contraint $(1)$ is indeed lifted.
Since the observables are $x^i$ in terms of $x^0$, the parametrization of the trajectory is irrelevant. The action and the equations of motion should be invariant under reparametrization. It is easier to work out the Euler-Lagrange equation with an arbitrary parametrization, and choose a sensible one (proper time) at the end.
