# Kepler problem: flows generated by constants of motion

1. Aug 30, 2012

### Clausius

Consider the Hamiltonian of Kepler problem
$$H(\boldsymbol{r},\boldsymbol{p})= \frac{|\boldsymbol{p}^2|}{2\mu} +\frac{\alpha}{|\boldsymbol{r}|}, \qquad \mu>0>\alpha,$$
where $$\boldsymbol{r}\in M=\mathbb{R}^3\setminus\{ 0 \}, \ (\boldsymbol{r},\boldsymbol{p})\in T^*M$$ and
$$|\boldsymbol{r}|=\sqrt{r_1^2+r_2^2+r_3^2}.$$
The quantities
$$\boldsymbol{m}=\boldsymbol{r}\times\boldsymbol{p}, \qquad \boldsymbol{W}=\boldsymbol{p}\times\boldsymbol{m}+\mu\alpha\frac{\boldsymbol{r}}{|\boldsymbol{r}|}$$
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 $$\Phi: T^*M\to T^*M$$ is a point transformation if it is induced by a transformation $$\phi:M\to M,$$
so that
$$\Phi(\boldsymbol{r},\boldsymbol{p})= (\phi(\boldsymbol{r}),\phi^{*-1}), \ \phi^*_i= \frac{\partial\phi_i}{\partial r_j}p_j.$$