Why does the given conserved quantity mean the motion is on a cone?

[deuteron]

TL;DR Summary: .

An electrone moves in a magnetic field $B(\vec r)=g \frac {\vec r}{|\vec r|^3}$. Why does the conservation of the quantity $$\vec J=\vec r \times\vec p +eg\frac {\vec r}{|\vec r|}$$ mean that the motion is on the surface of a cone?

 

[PeroK]

Is this homework?

 

[vanhees71]

You multiplying $\vec{J}$ with $\vec{r}$ gives
$$\vec{r} \cdot \vec{J}=e g r.$$
Now use spherical coordinates with $\vec{J}/J$ as the polar axis. Then the equation implies
$$J x_3 =e g r \; \Rightarrow \; \cos \vartheta=\frac{x_3}{r}=\frac{e g}{J}=\text{const},$$
which is the (implicit equation of a cone).

In the spherical coordinates you thus have
$$\vec{r}=\begin{pmatrix} r \sin \vartheta \cos \varphi \\ r \sin \vartheta \sin \varphi \\ e g r/J \end{pmatrix},$$
which describes a cone since $\vartheta=\text{const}$.