Let $(M,\omega)$ be a compact symplectic manifold. For any $H\in C^{\infty}([0,1]\times M)$, the Hamiltonian vector field $X_t$ on $M$ is defined by \begin{equation}dH_t=\iota_{X_t}\omega.\end{equation}(Here $H_t(x):=H(t,x)$)
$X_t$ flows a (time dependent) 1-parameter family of diffeomorphism $\phi_t^H$. $\phi$ can be seen as a map $M\times C^{\infty}([0,1]\times M)\to M$ by \begin{equation}(x,H)\mapsto\phi^H_1(x)\end{equation}
Let $x\in M$, $H\in C^{\infty}([0,1]\times M)$. Take the linearization of the above map, consider the map $C^{\infty}([0,1]\times M)\to T_{\phi^H_1(x)}M$ \begin{equation}h\mapsto D\phi^H_1(x)(h,0).\end{equation}
I want to prove the above map is surjective.
My attempt: Let $v\in T_{\phi^H_1(x)}M$, $h\in C^{\infty}([0,1]\times M)$, and $Y_t$ be the Hamiltonian vector field of $h$. I guess $Y_1(\phi^H_1(x))=v$ is enough to guarantee $D\phi^H_1(x)(h,0)=v$. I know that I should calculate \begin{equation}D\phi^H_1(x)(h,0)=\dfrac{d}{dr}\Big|_{r=0}\phi^{H+rh}_1(x).\end{equation} I've tried to differentiate \begin{equation}\dfrac{d}{dt}\phi^{H+rh}_t(x)=(X_t+rY_t)(\phi^{H+rh}_t(x))\end{equation}both sides w.r.t $r$, but it turns out to be a mess.
Background: Originally I would like to prove that for generic Hamiltonian diffeomorphism $\phi$ and Lagragian submanifolds $L_0$ and $L_1$, $L_0$ intersects $\phi(L_1)$ transversely. If I get the surjectivity, I can use implicit function theorem. After that I can proceed using Sard-Smale's theorem.
I do not understand how you plan to get transversality from what you said. (I do not claim here that you are mistaken, but only that I do not see how the argument is supposed to work.) So I will content to make a few comments.
1) For what you aim to do, observe that you can always renormalize $H(t,x)$ to $F(t, x) := a'(t)H(a(t), x)$ where $a : [0,1] \to [0,1]$ is any smooth increasing bijection. You then have $\phi^F_t = \phi^{H}_{a(t)}$ and so in particular $\phi^F_1 = \phi^{H}_1$. You can choose $a$ such that $a \equiv 1$ on some interval $(1- \epsilon, 1]$.
It is somewhat easier to answer 'what could be the end result of perturbing $F$' than to answer the corresponding question with $H$, as $F$ acts trivially near time 1. For instance, you can perturb using a function $h$ with 'time-support' in $(1- \epsilon, 1]$; You can even take $h(t,x) = b(t)h(x)$ with $b$ with support in $(1- \epsilon, 1]$ and $X_h(x) = v$. The flow of $(F+h)(t,x)$ is then the mere concatenation of the respective flows of $F$ and $h$. Among those perturbations, it is not difficult to see that your guess is correct. (Your guess might be correct also in general. I reckon Banyaga considers the effect of smoothly varying an Hamiltonian function in his thesis.)
2) The result holds at any given point $x$. Whichever plan you come up with to prove transversality, it would have to be of a global nature and not just of a pointwise one. For instance, if $H \equiv 0$, $L_0 = L_1$; To perturb $L_1$ into $L'_1$ which is transverse to $L_0$, you cannot only perturb $L_1$ at one point $L_0 \cap L_1 = L_0$. It is not even clear if a succession of pointwise perturbations would suffice.
3) I guess the safest way to prove that $L_1$ could be perturbed into a Lagrangian $L'_1$ transverse to $L_0$ would be to mimic the proof in the merely smooth context. The philosophy comes more or less from Gromov's h-principle: If you have a (connected) group like $Diff_0(M)$ or $Ham(M, \omega)$ which can move the manifold a lot even on very small scales (both groups are, for instance, $n$-transitive for any integer $n$), then you can use an isotopy of such transformations (starting at the identity) to bring a given submanifold into a transversal for another given one (modulo some hypotheses on their (co)dimension).
For your purposes, this strategy works well in combination with (1). However, for different purposes, you might really want to perturb $H$ with a function $h$ without constraints on the 'time support' of $h$. Observe however that given a Lagrangian embedded submanifold $L \subset (M, \omega)$ and a time-dependent Hamiltonian function $H$ on $(M, \omega)$, the map $J_H : L \times [0,1] \to (M \times [0,1] \times \mathbb{R}, \omega \oplus dt \wedge dE) : (x,t) \mapsto \left(\phi^{H}_t(x), t, H(t, \phi^{H}_t(x)) \right)$ is a Lagrangian embedding (I might be missing some signs here, but that is essentially the form of the embedding). Therefore, you might be able with the help of a Hamiltonian flow on $M \times [0,1] \times \mathbb{R}$ to $C^1$-perturb (relatively to $L \times \{0\}$) this Lagrangian embedding into another one which would be transverse to it (and which would be at time $1$ transverse to $\phi^{H}_1(L)$). This new Lagrangian would be the image of some $J_{H+h}$, giving you the perturbation $h$.