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I am currently reading this paper on symplectic geometry.

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 $d\hat{F}^i(\delta x_0).$

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This question was asked and answered on MathOverflow. I have replicated the accepted answer by user17945 below.

It just looks like a basic application of the chain rule to the immediately preceding equation. Maybe it's the notation that's confusing you; the previous equation has the form $$ (\partial_2F)(x,y) + (\partial_1 G)(y,z) = 0. $$ Considering the left hand side as a function of three independent variables $(x,y,z)$, and differentiating in the direction $(\delta x, \delta y, \delta z)$, gives $$ \partial_1\partial_2 F\, \delta x + (\partial_2\partial_2F + \partial_1\partial_1G)\,\delta y + \partial_2\partial_1 G\,\delta z = 0, $$ which is equation $(A.2)$. This gives a constraint that must be satisfied by the three variations $(\delta x, \delta y, \delta z)$.

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