Kepler problem: flows generated by constants of motion

[Clausius]

Consider the Hamiltonian of Kepler problem
[tex] H(\boldsymbol{r},\boldsymbol{p})=
\frac{|\boldsymbol{p}^2|}{2\mu}
+\frac{\alpha}{|\boldsymbol{r}|},
\qquad \mu>0>\alpha,[/tex]
where [tex]\boldsymbol{r}\in M=\mathbb{R}^3\setminus\{ 0 \}, \ (\boldsymbol{r},\boldsymbol{p})\in T^*M[/tex] and
[tex]|\boldsymbol{r}|=\sqrt{r_1^2+r_2^2+r_3^2}.[/tex]
The quantities
[tex] \boldsymbol{m}=\boldsymbol{r}\times\boldsymbol{p},
\qquad \boldsymbol{W}=\boldsymbol{p}\times\boldsymbol{m}+\mu\alpha\frac{\boldsymbol{r}}{|\boldsymbol{r}|}[/tex]
are constants of motion, as is well known.

My question is: how can I prove that the flows generated by the functions m_i and W_i, i=1,2,3 are canonical transformations?
Moreover, are such transformations point transformations?

A canonical transformation [tex]\Phi: T^*M\to T^*M[/tex] is a point transformation if it is induced by a transformation [tex]\phi:M\to M,[/tex]
so that
[tex]\Phi(\boldsymbol{r},\boldsymbol{p})=
(\phi(\boldsymbol{r}),\phi^{*-1}),
\ \phi^*_i=
\frac{\partial\phi_i}{\partial r_j}p_j.[/tex]

 

[LightHero]

In order to prove that the flows generated by the functions m_i and W_i are canonical transformations, we must first determine the generating functions of these flows. Generating functions of the form S(\boldsymbol{r},\boldsymbol{q}) can be used to define canonical transformations through the equations \boldsymbol{p}=\frac{\partial S}{\partial\boldsymbol{q}},\ \ \boldsymbol{r}=\frac{\partial S}{\partial\boldsymbol{p}}.We then use Hamilton's equations\frac{d\boldsymbol{r}}{dt}=\frac{\partial H}{\partial\boldsymbol{p}}and\frac{d\boldsymbol{p}}{dt}=-\frac{\partial H}{\partial\boldsymbol{r}}to prove that the flows generated by the functions m_i and W_i do indeed generate canonical transformations.To show that the transformations are point transformations, we must additionally prove that there exists a transformation \phi:M\to M such that\Phi(\boldsymbol{r},\boldsymbol{p})=(\phi(\boldsymbol{r}),\phi^{*-1}),\ \phi^*_i=\frac{\partial\phi_i}{\partial r_j}p_j.This can be established by solving the Hamilton-Jacobi equation for the generating functions S(\boldsymbol{r},\boldsymbol{q}) and then constructing the corresponding transformation \phi:M\to M from the solution.