1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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})=
    \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
    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
    \ \phi^*_i=
    \frac{\partial\phi_i}{\partial r_j}p_j.[/tex]
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted