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Kepler problem: flows generated by constants of motion

  1. Aug 30, 2012 #1
    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]
     
  2. jcsd
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