\dot{r}

. The magnetic cross-product term is perpendicular to this velocity, so

m \dot{r} \cdot \ddot{r} = - k r \cdot \dot{r} - e \dot{r} \cdot (\dot{r} \times B) = - k r \cdot \dot{r} .

Thus

\frac{d}{d t} (m ∣ \dot{r} ∣^{2} + k ∣ r ∣^{2}) = 0,

\frac{1}{2} m ∣ \dot{r} ∣^{2} + \frac{1}{2} k ∣ r ∣^{2} = constant .

The two halves are the kinetic energy and the isotropic spring's potential energy. The expression in the question is twice their sum. The Lorentz force changes the direction of motion but supplies no power, because it is perpendicular to velocity; this explains why the magnetic field does not appear in the conserved mechanical energy. The sign corresponds to charge