Consider the Hamiltonian of Kepler problem(adsbygoogle = window.adsbygoogle || []).push({});

[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]

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# Homework Help: Kepler problem: flows generated by constants of motion

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