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Suppose I have a particle under the influence of an electric field. The particle is also attached to a spring. The particle should therefore feel the influence of two forces such that the total force is then:

\begin{align} F_T &= F_E + F_S\\ &= qE - kx \end{align} where $k$ is just the spring constant. This is then a second order differential equation:

$$ m\frac{\mathrm d^2x}{\mathrm dt^2} = qE - kx $$

What I am struggling with is solving this equation such that I get an expression for $\Delta x$ which is essentially the distance moved by the particle in some interval $\Delta t$.

For context, the spring is a rough model for the surface tension of a surrounding material pulling the particle back toward the equilibrium point of the surface. This is going into a computer model, with arbitrary time-steps, with each step calculating the change in $x$, $\Delta x$. To be clear, the potential applied to the particle is constant, but the electric field depends upon $x$. So if we say that the particle is being attracted toward some source, the field strength and therefore the force will increase as the particle moves closer to the source.

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  • $\begingroup$ Is this homework? If so, then please add the homework-and-exercises tag. $\endgroup$ – Ben Crowell Dec 5 '19 at 21:59
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The general trick is to convert the one second-order differential into two first-order ones. This is done by introducing an extra variable to carry around: \begin{align} \frac{\mathrm dv}{\mathrm dt}&=f(x,\,t)\\ \frac{\mathrm dx}{\mathrm dt}&=v \end{align} for whatever function $f(\cdot)$ you need.

This can easily be applied to numerical integration schemes (Runge-Kutta, verlet, etc) for which you seem to be interested in.

Given the electric field under consideration here, you will likely end up with a current being generated, and hence have the magnetic field play a role (though perhaps you could argue for not including it). In either event, you be interested in particle in cell method (has been discussed on this site as well).

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