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On the surface, it a straight forward physics problem and I was only looking at it to learn how to use Mathematica. At first I thought it was noise, but now I am not so sure. If I use Plot and use the separation distance as the variable, I get some strange discontinuities at certain distances and sizes. It could be a scattering problem if I were looking at the actual dynamics as I change the separation distances. Even though the problem is a static problem, I am loath to simply throw the imaginary components away because it is possible that there are some phase effects that the integrals are showing me. Now, as I fool around with the numerical integration making the sizes and distances of the spheres different, these imaginary components to the solutions begin to show up.not always but under certain circumstances. Stories involve gliders, fighter jets, private aircraft, top-dressing planes, helicopters and many military aircraft, including the famous F-18 from. Using spheres I also do not have additional relative orientation geometries to make thing even more intractable. The pilots represent a range of flying eras, aircraft and experiences - from the Second World War to the present day, from one of the original Dambusters to flying Richard Bransons private jet. If I further modify the geometry from disks to spheres using cylindrical symmetry, the integrand is only a little more complex and the same limits on finding an analytical solution exist. I switched to numerical integration so long ago I forget how I got there!!! If I extend the ring into a disk, the resulting integral is similar to an elliptical integral but cannot be solved analytically because now the denominator of the integrand contains additional spatial variables, at least if am correctly recalling my efforts at having Mathematica solve that integral. If I replace the test charge with another coaxial ring, my result is an elliptical integral. If the test charge is off-axis, the mutual force requires the solution to an elliptical integral. If the test charge is on the ring axis, the solution is trivial.
If I put a test charge in the vicinity of the first ring, I find a mutual force between the ring and the test charge. Take a charged ring.so much charge per unit area of the ring. They may mean something that I simply cannot throw away or at least I do so at some risk to the validity of my results.
But the numerical integration works very nicely, except that over a certain limited range and size of variable these pesky imaginary components spring up. My integrals are similar to Elliptical integrals, but there are additional terms in the denominator of the integrand function that make an analytical solution impossible.
On the other hand, I remember some old physics scattering problems in which the imaginary parts represented phase shifts that caused the output wave functions to oscillate with interference patterns.and these were measured and seemed to be real effects. My take away is that I CAN ignore the imaginary component, even if it is very large. I relied on the smart PhDs working for me to get it right!!! LOL. Once I went into management my brain atrophied. Again in the distant past my Fortran integrals using the simple numerical models and were all in the real domain. My experience in the complex domain was limited to various engineering discussions and formal methods such as Laplace transforms. Finance, Statistics & Business Analysis.Wolfram Data Framework Semantic framework for real-world data.
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