I would like to check my global understanding on G.R: how we determine the trajectories of particles.

Everything starts from the equations of Einstein. 

We have to find a metric that satisfies thoose equation: this is the first step (which is generally really hard).

Once it is done, we want to find the trajectories of particles. We know that thoose trajectories will be geodesics. It mean that they will minimise the length:

$$ L=\int ds = \int \sqrt{g_{\mu, \nu} \frac{dx^{\mu}}{dt} \frac{dx^{\nu}}{dt}} dt.$$

**Little question**: how do we know that the particles will follow geodesics? Is it a postulate of GR?

To find the minimum of this quantity is equivalent as finding the minimum of:

$$ \int g_{\mu \nu} \frac{dx^{\mu}}{dt} \frac{dx^{\nu}}{dt} dt. $$

Thus, we can say that the Lagrangian describing our moving particle is in fact:

$$ \mathcal{L}=g_{\mu \nu} \frac{dx^{\mu}}{dt} \frac{dx^{\nu}}{dt}.$$

Of course this Lagrangian has "no link" with the lagrangian of GR. The last one describes the equation of the fields and lead to the einstein equation, whereas this one is the lagrangian of a moving particle in the theory of general relativity.

**Am I right with everything I  wrote?**

In fact I have a general relativity class this year which is supposed to be a part II and I haven't followed the part I so I need to catch up fast. This is why my questions are probably basics ones.