The same equations were used to generate this image as well. Â The only difference is the starting point. Â Instead of expanding from the center of the distortion field, the starting point is displaced a little bit towards the bottom. Â This highlights a number of different things. First while the equations areÂ symmetric, the resulting image is decidedly asymmetric. Â If you follow the lines to the right vs. the ones to the left, you can see that the paths have decidedly different curves.
These renderings never really went anywhere. They show chaotic orbits around 3 fixed attractive potentials while varying a uniform magnetic field. There are some lovely sweeps and folds as the chaos unfolds. I also like the glowing effect as the orbits get tightly wrapped around the centers of attraction. There wasn’t much to this other than some pretty curves, however, I did realize how essential drag is in making interesting forms. I suppose adding flocking and some higher order terms would help in adding further visual interest.
This is what you get if you take the parallel stream from the rest of the series and replace them with a circular spray from the center of the distortion. Â Everything gets bent uniformly when it hits the ring. Â Moving to second order equations gives you the freedom to have multiple directions from a single point like this without a singularity.
Double twist and wrap. Â While the last image was on the lackluster side, this one has everything turned up to 11. Â There is a strong deflection as the particles pass through the first peak and then they are swung back again as they exit the ring on the other side. Â An important thing to notice is that these are clearly defined and there is little mixing. Â If these equations were chaotic, then there would be mixing among the lines.
In my last few posts, I’ve been trying to characterize different potentials through the shapes of their orbits (gravitational, harmonic, lorentz). In the middle of it, I came across a post on bad astronomy about gravitational slingshots. I figured it would be a perfect opportunity to use these images to show the effect in a different way.
To start, I’m sending a beam of particles directly towards the planet in question, just like my previous demos. This time I slowly increase the speed of the planet and the orbit shapes change accordingly. It was pretty hard to see what was going on with the first few renderings, so I started by increasing the intensity and color of the orbit with the speed of the particle. Things were starting to look better, but they didn’t really start to illuminate the dynamics until I subtracted off the initial velocity of the test particles and only showed particles that were moving faster than their initial speed.
This one shows the default case of a stationary target. The particles accelerate as they get close and then slow down to their initial speed as they move a way.
This is the first shot of a moving target. You can see there are now a few particles that are being shot back at a higher speed, extending past the first line of acceleration from the last image.
There are a few more features visible now as the speed picks up. Since these are lorentz potentials, you can see a more pronounced central core of paths that pass right by the target and are only slightly deflected. The same thing happens with particles further out, but there is a sweet spot that generates two beams. The faster the target moves, the less deflection is seen. With a slow target, the final trajectory is almost a 180, but the angle decreases more and more as the target speed increases, though the final boost in speed increases as well.
I wanted to finish up with one image that captured the gravitational result. There are many similarities, but some significant differences. The infinity at dead center means that there isn’t a max deflection angle like there is in the lorentz case. As you you approach dead center, there will always be a set of bound states. This leads to simpler images, and really those trajectories aren’t all that interesting as far as the slingshot goes. Hyperbolic orbits are the only ones that get launched somewhere.
Anyway I hope these images illustrate some of the features of this process. I’ve learned a lot putting them together, both about these processes, and how to illustrate them in a way that makes the physics visible to the naked eye.
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.