A less idealized central force

This is the 3rd in a series that started with the gravitational potential, and then the harmonic. While those are the two biggies, what we are looking at next behaves more like something you would see in real life, and doesn’t have solutions that can be written down with simple formulas.

For this entry in our series, I reached into my hat and pulled out the lorentz distribution from basic physics. It it finite in all ranges from the very small, all the way out to infinity, and dare I say beyond.

These images range from a stream of slow particles which converge directly to the center of the force field, through a range of velocities, until the force is just a blip to be zoomed over, producing just a slight deflection.

Some features to note. The intermediate images have more features and details than either the harmonic or gravitational wells. That is directly related to the cut offs in the force. Only the harmonic and gravitational potentials have elliptical orbits, all other potentials have more complicated and complex shapes.

Eventually, however the streams get fast enough that there are no major deflections and no bound orbits. That just shows up as a narrow beam of deflected particles.

Central force images #2

This is the second in a series profiling the solutions to common central force fields from physics. Check out the the first one on the Newtonian gravitational force field.

These are even more plain than the gravitational well. That’s because these are for the harmonic potential. One of the things that distinguishes this potential is that all orbits are bound and elliptical in shape.

Contrasting that with the gravitational potential, there are 3 orbit shapes for that force: elliptical, parabolic, and hyperbolic. You can see the effects in the image. There is a relatively dark area bounded by the parabolic orbit, with a diffuse bright spot near the tip of the parabola. This is due to the tightly curved hyperbolic orbits nearby. The bright spot fades out into a dim patch further out as the trajectories become straighter the further they are from the center of the field.

Both of these fields have infinities which means that they valid only as approximations to actual forces. The harmonic field has the most severe, it extends out to infinity and no particle can escape from it’s pull. It doesn’t matter how fast our test particles are moving, they will never escape the grasp of the central pull.

Newtonian gravity has the opposite problem. The force becomes infinitely strong as you approach the central point. This causes problems for the differential equation solver and leads to the lower two images with kinked up trajectories and the rays radiating from the central point.

In our next installment I’ll be showing a field without those infinities and see how it stacks up.