# Particle motion with discrete values for the potential

Sorry if the question seems trivial, but I am new to physical simulations in Mathematica. I have a numerical estimation of an electric potential $V(x)$ in 3D, to be more precise I have a matrix containing $[x , y, z , V(x, y, z)]$ in a cubic domain, so the potential in not an analytic function . Is there a method, using Mathematica, to solve numerically the equation of motion for a charged particle in the potential? Essentially I would like to solve $\frac{d^2}{dt^2} \textbf{x}(t) = e \nabla V (\textbf{x}(t))$ with certain initial conditions.

• Is your discrete function a sampling or approximation of an underlying continuous function? If so, use FindFit to get the 3D potential and then Solve to calculate the particle trajectories. (If instead your potential is constant within a cubelet, then the derivatives vanish and if your particle ever sits within a cubelet, it will remain in that cubelet forever, even though regions with lower potential may exist.) Feb 20, 2015 at 19:20
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• No, the discrete function is the result of a finite element simulation. An analytic model for the potential doesn't seem to be available. Feb 20, 2015 at 19:28
• Please take a look at Interpolation[] Feb 20, 2015 at 19:42

For example:

(*Your matrix*)
max = 2 Pi;
pot = Table[{{x, y, z}, Sin@x y z } // N, {x, 0, max, max/40}, {y, 0, max,  max/20},
{z, 0, max, max/20}];
v = Interpolation@Flatten[pot, 2];
dx[x_, y_, z_] := D[v[r, y, z], r] /. r -> x
dy[x_, y_, z_] := D[v[x, r, z], r] /. r -> y
dz[x_, y_, z_] := D[v[x, y, r], r] /. r -> z

sol = NDSolve[{
x''[t] == dx[x[t], y[t], z[t]], x[0] == 1, x'[0] == 1,
y''[t] == dy[x[t], y[t], z[t]], y[0] == 1, y'[0] == -1,
z''[t] == dz[x[t], y[t], z[t]], z[0] == 1, z'[0] == 0
},
{x[t], y[t], z[t]}, {t, 0, 2}];

ParametricPlot3D[{x[t], y[t], z[t]} /. sol, {t, 0, 2}]


• Your solution seems to work correctly. The only issue is related to the mesh of the sampling that in my case is not uniform and gives a warning. The solution is computed though. Feb 21, 2015 at 17:47
• @user47224 If you search this site you'll find a couple of questions about interpolating in non-uniform grids. Thanks for the accept! Feb 21, 2015 at 17:55