Let $(M,\omega)$ be a (symplectic) manifold.
I want to compute the Maslov index of a loop $\gamma:\mathbb{R}\to M$ directly. In order to do that I have to find a (symplectic) trivialization of $\gamma^*TM$ but I can't see how to do this in general. Many references say to use a (symplectic) trivialization of $u^*TM$ for $u$ a map from the disk to $M$ agreeing with $\gamma$ on the boundary. Here below what I've done.
The example I tried to work out is $M=S^2$ and $\gamma(t)=(\cos(t),\sin(t),0)$. The $u$ one could consider is $$ u: D\to S^2\\ (x,y)\mapsto (x,y,\sqrt{1-x^2-y^2}) $$ In this case, as $M$ is two dimensional I would be tempted to use a coordinate chart $\chi:U\to \mathbb{R}^2$ (e.g. stereographic projection from south pole), which would give a map $$ D\times \mathbb{R}^2\to u^*TM\\ $$ but this works only for $\dim M =2$ so it's probably the wrong approach.
It would be very helpful to have (a reference to) an example of such a concrete computation.
Just typing out the details of Ted's comment.
The solutions follows directly from $\gamma$ (or $u$) being contractible. This is a simple application of parallel transport as might be checked for example here. What one receives is a map from a fix $q\in im(u)$ to any $p\in im(u)$ $$ T_q M \to T_p M $$ which thus is a global frame on the pullback bundle $u^*TM$. The symplectic bit follows from separate computation.