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Suppose I have a charged particle $q$ with mass $m$, and an infinite long wire that lies on the $z$ axis, in which flows a constant current $I$. My ODE that describes its motion is

$$\vec {\ddot r}=\dfrac{q}{m}\vec {\dot r} \times \vec B$$ where $\vec B$ is the wire's tangential magnetic field it makes. Is there any importance for the plot of $({\vec {\dot r(t)}})^2$ as a function of time in understanding the solution of the problem, given two initial values for $\vec r(t)$ and $\vec {\dot r}(t)$? I am solving a numerical exercise and being asked to plot this graph. Any suggestions?

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    $\begingroup$ I suppose plotting $\dot r^2$ could give you a rough visual representation of how the particle's kinetic energy behaves $\endgroup$
    – Jim
    Commented Jul 14, 2015 at 16:37

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I think the point of the exercise is to show you that $(\dot{\vec{r}})^2$ is constant.

Since magnetic fields do no work (equivalently, since the acceleration a magnetic field produces is always perpendicular to velocity), the magnitude of the velocity never changes. So, by plotting $(\dot{\vec{r}})^2$ and making sure it's constant, you can check if you made a mistake in finding the solution for $\vec{r}$.

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  • $\begingroup$ That sound right. I must agree with you. $\endgroup$
    – E Be
    Commented Jul 14, 2015 at 18:00

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