How can I show (using the Euler Lagrange Equations) that : for constants (mass and time) $m,h>0$
\begin{align} h\inf \bigg\{ &\int_0^h m\left\| \frac{d^2}{dt^2}\xi (t) \right\|^2 : \xi \in C^1([0,h];\mathbb{R}^d), \\ &(\xi,m\dot{\xi})(0)=(q,p), \ (\xi,m\dot{\xi})(h)=(q',p') \bigg\} \\ =&\|p'-p \|^2 + 12\left\| \frac{m}{h}(q'-q)-\frac{p'-p}{2} \right\|^2. \end{align}
Im confused how to apply these equations.
Using the Euler Lagrange equation, you can show that the infinimum is indeed reached for a function $\varphi \in C^1([0,h],\mathbb{R}^d)$ that verifies :
$$ \frac{d^4 \varphi}{dt^4}(t)=0, \quad \forall t \in [0,h]\\ \varphi(0)=q\\ \varphi'(0)=\frac{p}{m}\\ \varphi(h)=q'\\ \varphi'(h)=\frac{p'}{m} $$
The first equation gives you that there exists constants $(c_3,c_2,c_1,c_0)\in (\mathbb{R}^d)^4$ such as : $$\varphi(t)=c_3t^3 +c_2 t^2 + c_1t+ c_0$$ and you find the four constants using the 4 boundary conditions.
After that all you need to compute is :
$$h m \int_0^h ||\underbrace{6 c_3 t+2 c_2}_{\frac{d^2}{dt^2}\varphi(t)}||^2 \ \mathrm{d}t$$ which should give you the good formula.