Consider the Lagrangian $L:\mathbb{R}^2\times \mathbb{R}^2\to \mathbb{R}$
$$ L(\sigma,\dot{\sigma}):=\Big\langle \begin{pmatrix} \frac{1}{h} & 0 \\ 0 & 1 \end{pmatrix}\dot{\sigma}+ \begin{pmatrix} 0 & \frac{1}{h} \\ 0 & 0 \end{pmatrix}\sigma,\dot{\sigma} \Big\rangle $$
Note we know the boundary values as $\sigma(0)=(q,p)$ and $\sigma(1)=(q',p')$. I calculated the Euler Lagrange equations as
$$ \begin{pmatrix} 0 & -\frac{1}{h} \\ \frac{1}{h} & 0 \end{pmatrix}\dot{\sigma} -2\begin{pmatrix} \frac{1}{h} & 0 \\ 0 & 1 \end{pmatrix}\ddot{\sigma}=0.$$
From here how do I access $\sigma$? can I solve these equations? If so how?