I am currently reading a paper on symplectic geometry: Periodic orbits for Hamiltonian systems in cotangent bundles by Christopher Golé.

It deals with the question how the stability properties of a sequence of (periodic) points or fixed points can be related to the second derivative of the action functional. So the underlying question that is covered in this theorem is how stability/instability of a point relates to minimizers/maximizers of the action functional.

The proof of this is partly done in the appendix on p.22, but I don't really see how equation A.2 is derived.

So the situation is something like this (it is a reduced situation of the original question):

We are given a symplectomorphism $F$ with a fixed point $x = (q_0,p_0).$ In this situation, this means that $\hat{F}^{2N}(x)=x.$ Notice, that $F$ here is actually a composition of $2N$ many maps. (see page 10)

So, we can think of $F$ as being $F= \hat{F} \circ...\circ \hat{F}$ $2N$ times, where $\hat{F}$ is a symplectomorphism.

Furthermore, $(q_i,p_i)_i$ is an orbit under $\hat{F},$ so $\hat{F}^i(q_0,p_0) = (q_i,p_i).$Thus, the Euler-Lagrange equation is satisfied which is here the equation between $A.1$ and $A.2$.

What I now really don't understand is, how they get equation $A.2$?

If you have any questions, please let me know.

Does anybody see where it comes from? Maybe I am also misinterpreting their notion of a tangent orbit, but to me it is just an arbitrary tangent vector $\delta x_0 := (\delta q_0, \delta p_0)$ and then they iterate $d\hat{F}^i(\delta x_0).$