The Lagrangian of fractional classical mechanics $L_\alpha(\dot{q},q)$ is defined as
$$
L_\alpha(\dot{q}, q)=p\dot{q}-H_\alpha(p,q)
$$
where
$$
\begin{split}
p & =\frac{\partial L_\alpha(\dot{q}, q)}{\partial \dot{q}}, \\
H_\alpha(p, q)&=D_\alpha |p|^\alpha + V(q), \qquad 1 < \alpha \le 2
\end{split}
$$
How to obtain the following using above definitions?
$$
L_\alpha(\dot{q}, q)=\left(\frac{1}{\alpha D_\alpha}\right)^{\frac{1}{\alpha-1}} \frac{\alpha-1}{\alpha}|\dot{q}|^{\frac{\alpha}{\alpha-1}}, \qquad 1 < \alpha \le 2
$$

**Source:** *Fractional Quantum Mechanics*, Nick Laskin, World Scientific, 2018, page 260, [MR3821542](https://mathscinet.ams.org/mathscinet-getitem?mr=3821542), [Zbl 1425.81007](https://zbmath.org/1425.81007).