How to obtain the Lagrangian of the fractional classical mechanics

Question

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, Zbl 1425.81007.

Answer 1

How to obtain the following using above definitions?

Referring to this note by Nick Laskin:

1. Get $|p|$ from the first of Eqations (15);

2. Substitute this $|p|$ into Eq. (4) assuming $V(q)=0$ in the definition of $H(p,q)$ given by Eq.(2);

3. After trivial algebra we obtain Eq. (7), which is the equation in question.