How is proper time extremized? I just completed an exercise that asked me to prove that, in special relativity, free particles move with uniform velocity on geodesics that are straight lines. After doing this problem, I was wondering why one should choose to extremize proper time when doing this calculation?
In section 5.3 of Guidry (titled Geodesics and a Variational Principle), it says "the worldline for free particles between points separated by timelike intervals extremizes the proper time". I am wondering why this is interesting or what more there is to this?
I have heard that in mechanics what is extremized is the proper time, meaning things do operate on this principle and will move in the way that extremizes proper time. My question is how or why this happens. My professors love to tell me there is no answer to "why", but surely there is an answer to "how", or at least something deeper here?
 A: First off, an answer to the question "why do timelike geodesics extremize proper time".
A geodesic is by definition the path with the shortest distance between two points on a manifold i.e. you have the minimum of the proper distance:
\begin{equation}
\Delta s = \int ds
\end{equation}
where $ds^{2}$ the infinitesimal interval that will describe the metric of your manifold. We can equivalently define $d\tau ^{2} = -ds^{2}$ and thus, if we have a timelike interval ($ds^{2} < 0$) we may define proper time as:
\begin{equation}
\Delta \tau = \int d\tau
\end{equation}
Minimizing the "distance" on the manifold ends up maximizing proper time.
Now to the second question, "why do free particles move along geodesics".
The relativistic Lagrangian for a free particle of mass $m$ is going to be:
\begin{equation}
L = mc^{2} \frac{d\tau}{dt}
\end{equation}
Thus if we write down the action:
\begin{equation}
S = \int dt \, L = mc^{2} \int d\tau
\end{equation}
Therefore the need to extremize proper time and hence for free particles to move along geodesics is tantamount to the principle of stationary/least action $\delta S = 0$.
Then the question could become "why do particles (or anything for that matter) need to obey dynamics so that the principle of least action holds", to which no one really has a definitive answer. One possible idea is to consider a fundamental substrate of quantum mechanics permeating classical mechanics via the path integral formulation. Transitional probability amplitudes are given by path integrals weighted by the action of the system:
\begin{equation}
\langle \, x_{f} , t_{f} \, | \, x_{i}, t_{i} \, \rangle = \int \mathcal D x(t) \, e^{\frac{iS[x(t)]}{\hbar} }
\end{equation}
We could then treat this quantity as a partition function and, after a Wick rotation $t \rightarrow \tau = it$, they will have the form:
\begin{equation}
Z = \int \mathcal D x(\tau) \, e^{\frac{-I[x(\tau)]}{\hbar} }
\end{equation}
where $I[x(\tau)]$ the positive-definite Euclidean action following the Wick rotation. Now it has an analogous form to that of the partition function in statistical mechanics. Much like in statistical mechanics statistical quantities are weighted by the total energy of their configurations, in quantum mechanics probabilities and expected values of physical quantities are weighted by their action. Extremizing the action $S$ minimizes the Euclidean action $I$ which then provides the highest statistical weight.
A: In flat spacetime in Cartesian coordinates, it is not hard to prove that given two events $\mathbf p_1 = (ct_1,x_1,y_1,z_1)$ and $\mathbf p_2\equiv (ct_2,x_2,y_2,z_2)$ which are timelike-separated (i.e. $c^2 \Delta t^2 -\Delta x^2 -\Delta y^2 - \Delta z^2 >0$ ), the curve connecting them which maximizes the proper time
$$\Delta \tau = \int_{t_1}^{t_2} \sqrt{1 - v^2} \mathrm dt, \qquad v^2 \equiv \dot x(t)^2+\dot y(t)^2+\dot z(t)^2$$
is the straight line
$$\mathbf x(t) = \mathbf p_1 + \left(\frac{\mathbf p_2-\mathbf p_1}{t_2-t_1}\right) (t-t_1).$$
This is a straightforward application of the calculus of variations. On the other hand, the fact that free particles observed from an inertial reference frame follow straight lines at constant speed is a postulate of the theory (which is well-supported by careful observation).  Putting those two pieces of information together, we can say the following:

In an inertial reference frame, free particles move along straight lines at constant speed timelike worldlines which maximize $\Delta \tau$ (along any two points on the worldline).

In Cartesian coordinates and in flat spacetime, the fact that straight worldlines maximize the proper time between two events is exactly analogous to how in flat space, straight lines minimize the distance between two points, with the difference (maximize vs. minimize) being due to the minus sign in the definition of $\Delta \tau$. Both facts are examples which show that we can sometimes relate local information (e.g. each step along the worldline is parallel to the last) to global information (e.g. the worldline as a whole maximizes the proper time between its endpoints).  From a certain viewpoint, this is precisely the power of calculus - relating local information about a function to global information.


I am wondering why this is interesting or what more there is to this?

One reason this fact is important is that in curved spacetime, exactly the same principle applies. It's not necessarily obvious what the generalization of straight lines should be, but the generalization of $\Delta \tau$ is much more straightforward.  As a result, we can say that in GR (just as in SR), free particles travel along worldlines which extremize $\Delta \tau$.
A: This is a case where if we generalize from special relativity to general relativity, the possible options become fewer: We want to find the trajectory of a massive point particles that moves between 2  spacetime events/points $P$ and $Q$ on a Lorentzian manifold $(M,g)$. We know from Einstein's principle of general covariance, the trajectory should be geometric, i.e. not depend on coordinate system. The only invariant at our disposal is the arclength (aka. proper time, cf. this Phys.SE post). It is natural to guess that the actual trajectory is extremizing the arclength, i.e. it is a geodesic. This turns out to be correct. See e.g. this Phys.SE post for an action principle.
A: 
How is proper time extremized?

"How" is easy. The proper time interval is:
$$
S = \int d\tau
$$
If the particle has a trajectory $x(t)$, going from $x_1$ at $t_1$ to $x_2$ at $t_2$, then the quantity to extremize is:
$$
S = \int_{t_1}^{t_2} dt \sqrt{1-\dot x^2/c^2}\;,
$$
which is extremized by the trajectory
$$
\ddot x_e = 0\;,
$$
Or, integrating twice:
$$
x_e(t) = \frac{x_2}{t_2 - t_1}(t - t_1) + \frac{x_1}{t_1-t_2}(t-t_2)
$$
