Accelerated Classical Atom

This is an animation of a pair of particles: a nucleus with a charge $+1$ and an electron with a charge $-1$. Initially, the nucleus is fixed, and the electron is in a circular orbit around it. Then the nucleus starts accelerating at a constant rate (that is, it's momentum increases as $p_n=(Ft,0,0)$).

The position of the nucleus solves $$ \dot r_n=\frac{p_n}{\sqrt{m_n^2+p_n^2}}=\frac{(Ft,0,0)}{\sqrt{m_n^2+F^2t^2}} $$ that is $$ r_n(t)=\frac1F\sqrt{m_n^2+F^2t^2}-\frac{m_n}F . $$

The electron is subjected to the electromagnetic force due to the nucleus. The electric and magnetic fields at position $r_n(t)+\rho$ are $$ E(\rho)=\frac 1{s^3}\left(\xi-|\xi|\dot r_n(\tau))(1-\dot r_n(\tau)^2)+(\xi-|\xi|\dot r_n(\tau))\times \ddot r_n(\tau)\right) $$ $$ B(\rho)=\xi\times E/|\xi| $$ with $$ \tau:=t-|\xi| ,\quad \xi:=\rho+r_n(t)-r_n(\tau) ,\quad s:=|\xi|-\xi\cdot\dot r_n(\tau) . $$ The position of the electron is $r_e(t)$, and the relative position is $\rho_e(t):=r_e(t)-r_n(t)$, which satisfies $$ \dot p_e=-E(\rho_e)-(\dot \rho_e+\dot r_n(t))\times B(\rho_e) ,\quad \dot \rho_e=\frac{p_e}{\sqrt{m_e^2+p_e^2}}-\dot r_n . $$

Note that the equations for $\tau$ and $\xi$ are coupled. One can avoid this by considering $\tau$ as an extra degree of freedom, which satisfies the differential equation $$ \dot\tau=1-\frac\xi s\cdot(\dot\rho_e+\dot r_n(t)-\dot r_n(\tau)) . $$

There are two plots below: the first shows the x,y axes, and the second shows the z,y axes.

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