- #1
Clausius
- 4
- 0
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]
[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]