if have problems getting my head around the following claim made by Moser in "Monotone twist mappings and the calculus of variations" and Gole in "Symplectic twist maps".
Setting:
Let $F : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the lift of a twist map on the cylinder, i.e. $F$ is area-preserving, has zero flux and $\partial_yX > 0$ if we write $F(x,y) = (X(x,y),Y(x,y))$.
Such a twist map comes with a generating function $S: \mathbb{R}^2 \rightarrow \mathbb{R}, (x,X) \rightarrow S(x,X)$ with the following condition on its derivative $dS = YdX - ydx$.
Let $L : \mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}$ be a time-dependent Lagrangian on $\mathbb{R}$, such that the critical points of the action are given by straight lines $\gamma: t \mapsto x_0 + t(x_1 - x_0)$ for $t \in [0,1]$. Furthermore we require that the action of such lines is equal to the generating function of $F$, evaluated on the endpoints, i.e.
$S(x_0,x_1) = \int_0^1 L(t,x_0 + t(x_1-x_0), x_1-x_0) dt$.
Claim:
Moser and Gole claim that given such a lagrangian $L$, the time-1 map $\phi^H_1$ of the associated hamiltonian $H:\mathbb{R}^2 \rightarrow \mathbb{R}$ is equal to the twist map $F$.
Ideas:
Let me collect the ideas I've had so far. Since the critical points of the action are given by straight lines $\gamma$, i can explicitely write down the time-1 map $\phi^L_1$ of the Euler-Lagrange vector field on $\mathbb{R}^2$. Integral curves of the vector field are given by curves $t \mapsto (\gamma(t),\gamma'(t)) = (x_0 + t(x_1-x_0), x_1-x_0)$ and hence the time-1 map is given by $\phi^L_1(x_0,v) = (x_0 + v, v)$, where $v = x_1-x_0$. I know that at least in the autonomous case the flow of the associated hamiltonian $H$ is conjugated to $\phi^L$ via the Legendre-transform $\mathcal{L}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I don't know though if this still works in the time-dependent case. My hope was to calculate the time-1 map of $H$ explicitely and to see that this is the same as the map $F$ by calculating $F$ from its generating function $S$ via its derivatives.
Any help (or references to introductory material on time-dependent Lagrangian systems) is much much appreciated!