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

$$
p=\frac{\partial L_\alpha(\dot{q}, q)}{\partial \dot{q}}, \\
H_\alpha(p, q)=D_\alpha |p|^\alpha + V(q), \qquad 1 < \alpha \le 2
$$

How to obtain the following using above definitions?

$$
L_\alpha(\dot{q}, q)=(\frac{1}{\alpha D_\alpha})^{\frac{1}{\alpha-1}} \frac{\alpha-1}{\alpha}|\dot{q}|^{\frac{\alpha}{\alpha-1}}, \qquad 1 < \alpha \le 2
$$

**Source:** Fractional Quantum Mechanics, Nick Laskin, World Scientific, 2018, page 260.