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Numerical solutions of sixth order differential equations
An object moving in three dimensions is described by six variables: $x$, $y$, $z$, $v_x$, $v_y$, and $v_z$. If the force on the object is known, then the motion can be described by six first order differential equations,
$\large \frac{dx}{dt}=v_x$ $\large \frac{dv_x}{dt}=F_x(x,y,z,v_x,v_y,v_z,t)/m$
$\large \frac{dy}{dt}=v_y$ $\large \frac{dv_y}{dt}=F_y(x,y,z,v_x,v_y,v_z,t)/m$
$\large \frac{dz}{dt}=v_z$ $\large \frac{dv_z}{dt}=F_z(x,y,z,v_x,v_y,v_z,t)/m$
Here $F_x$, $F_y$, and $F_z$ are the three components of the force, $m$ is the mass, and $t$ is the time. The form below can be used to numerically integrate these equations for a total number of $N_{steps}$ steps using a step size of $\Delta t$. The force can be specified as a function of $x$, $y$, $z$, $v_x$, $v_y$, $v_z$ and $t$.
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