= Solution
Dot <Newton's second law> with $\dot{\boldsymbol r}$. The magnetic cross-product term is perpendicular to this <velocity>, so
$$
m\dot{\boldsymbol r}\cdot\ddot{\boldsymbol r}
=-k\boldsymbol r\cdot\dot{\boldsymbol r}
-e\dot{\boldsymbol r}\cdot(\dot{\boldsymbol r}\times\boldsymbol B)
=-k\boldsymbol r\cdot\dot{\boldsymbol r}.
$$
Thus
$$
\frac d{dt}\left(m|\dot{\boldsymbol r}|^2+k|\boldsymbol r|^2\right)=0,
$$
or
$$
\boxed{\frac12m|\dot{\boldsymbol r}|^2+\frac12k|\boldsymbol r|^2
=\mathrm{constant}.}
$$
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 $-e$.
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