Thank´s for your answer, now i get it. :)
Now i´m trying to apply Noether-Theorem.
At first i´d try to get the transformation in such a form:
\vec{r} \longmapsto \vec{r}+\vec{a}\cdot \vec{\psi}(\vec{r})
Therefore:
\vec{\psi}(\vec{r})=1-g\vec{r}^2+\underbrace{2g \vec{r}}
But the...
Seems like i do the same mistake every time i try it...
(\frac{d \vec{\tilde{r}}}{dt})^2=(\vec{\dot{r}}^2-2 \vec{a} g(\vec{r}\cdot \vec{\dot{r}})+2g \vec{\dot{r}}(-\vec{a}\cdot \vec{r})+2g\vec{r}(\vec{a}\cdot \vec{\dot{r}}))^2
=(\vec{\dot{r}}-2g \underbrace{(\vec{a}(\vec{r}\cdot...
Thanks for your answer.
(\frac{d \vec{\tilde{r}}}{dt})^2 should be at least something like \vec{\dot{r}}^2(1+4g(\vec{a}\cdot \vec{r}))
But i can´t get it into this form, especially the underbraced part of it makes me think there´s sth. wrong...
Homework Statement
Show that the Lagrangian
\mathcal{L}=\frac{m}{2}\vec{\dot{r}}^2 \, \frac{1}{(1+g \vec{r}^2)^2}
is invariant under the Transformation
\vec{r} \rightarrow \tilde{r}=\vec{r}+\vec{a}(1-g\vec{r}^2)+2g\vec{r}(\vec{r} \cdot \vec{a})
where b is a constant and \vec{a} are...