Yeah, ‘Y’ is a lousy letter in the Mathematics Glossary. I have a half-dozen mathematics books on the shelf by my computer. Some is semi-popular stuff like Richard Courant and Herbert Robbins’s What Is Mathematics? (the Ian Stewart revision). Some is fairly technical stuff, by which I mean Hidetoshi Nishimori’s Statistical Physics of Spin Glasses and Information Processing. There’s just no ‘Y’ terms in any of them worth anything. But I can rope something into the field. For example …
Yukawa Potential
When you as a physics undergraduate first take mechanics it’s mostly about very simple objects doing things according to one rule. The objects are usually these indivisible chunks. They’re either perfectly solid or they’re points, too tiny to have a surface area or volume that might mess things up. We draw them as circles or as blocks because they’re too hard to see on the paper or board otherwise. We spend a little time describing how they fall in a room. This lends itself to demonstrations in which the instructor drops a rubber ball. Then we go on to a mass on a spring hanging from the ceiling. Then to a mass on a spring hanging to another mass.
Then we go onto two things sliding on a surface and colliding, which would really lend itself to bouncing pool balls against one another. Instead we use smaller solid balls. Sometimes those “Newton’s Cradle” things with the five balls that dangle from wires and just barely touch each other. They give a good reason to start talking about vectors. I mean positional vectors, the ones that say “stuff moving this much in this direction”. Normal vectors, that is. Then we get into stars and planets and moons attracting each other by gravity. And then we get into the stuff that really needs calculus. The earlier stuff is helped by it, yes. It’s just by this point we can’t do without.
The “things colliding” and “balls dropped in a room” are the odd cases in this. Most of the interesting stuff in an introduction to mechanics course is about things attracting, or repelling, other things. And, particularly, they’re particles that interact by “central forces”. Their attraction or repulsion is along the line that connects the two particles. (Impossible for a force to do otherwise? Just wait until Intro to Mechanics II, when magnetism gets in the game. After that, somewhere in a fluid dynamics course, you’ll see how a vortex interacts with another vortex.) The potential energies for these all vary with distance between the points.
Yeah, they also depend on the mass, or charge, or some kind of strength-constant for the points. They also depend on some universal constant for the strength of the interacting force. But those are, well, constant. If you move the particles closer together or farther apart the potential changes just by how much you moved them, nothing else.
Particles hooked together by a spring have a potential that looks like 
The spring potential and the gravitational/electric potential have an interesting property. You can have “closed orbits” with a pair of them. You can set a particle orbiting another and, with time, get back to exactly the original positions and velocities. (Three or more particles you’re not guaranteed of anything.) The curious thing is this doesn’t always happen for potentials that look like “something or other times r to a power”. In fact, it never happens, except for the spring potential, the gravitational/electric potential, and — peculiarly — for the potential 
Ah, but does the potential have to be r-to-a-power? And here we see Dr Hideki Yukawa’s potential energy. Like these springs and gravitational/electric potentials, it varies only with the distance between particles. Its strength isn’t just the radius to a power, though. It uses a more complicated expression:
Here ‘K’ is a scaling constant for the strength of the whole force. It’s the kind of thing we have ‘G M’ for in the gravitational potential, or ‘k’ in the spring potential. The ‘b’ is a second kind of scaling. And that a kind of range. A range of what? It’ll help to look at this potential rewritten a little. It’s the same as 
Yukawa introduced this potential to physics in the 1930s. He was trying to model the forces which keep an atom’s nucleus together. It represents the potential we expect from particles that attract one another by exchanging some particles with a rest mass. This rest mass is hidden within that number ‘b’ there. If the rest mass is zero, the particles are exchanging something like light, and that’s just what we expect for the electric potential. For the gravitational potential … um. It’s complicated. It’s one of the reasons why we expect that gravitons, if they exist, have zero rest mass. But we don’t know that gravitons exist. We have a lot of trouble making theoretical gravitons and quantum mechanics work together. I’d rather be skeptical of the things until we need them.
Still, the Yukawa potential is an interesting mathematical creature even if we ignore its important role in modern physics. When I took my Introduction to Mechanics final one of the exam problems was deriving the equivalent of Kepler’s Laws of Motion for the Yukawa Potential. I thought then it was a brilliant problem. I still do. It struck me while writing this that I don’t remember whether it allows for closed orbits, except when b is zero. I’m a bit afraid to try to work out whether it does, lest I learn that I can’t follow the reasoning for that anymore. That would be a terrible thing to learn.
nebusresearch 
That’s an interesting one!! Re closed orbits: I just remember that there are only two potentials that will make sure that every bound orbit is closed: A quadratic (Hooke’s Law, a spring) and a gravitational 1/r potential. Other potentials can have closed orbits, but it depends on initial conditions.
Proofs usually make use of all the constants – energy, angular momentum – to be subsituted in the equations of motion (or the constants emerge from applying Langrange’s formalism) and angular momentum gives rise to an effective ‘add-on’ potential. Then different substitutions are applied that better fit the geometry of the problem, like using 1/r rather than r and angles or polar coordinates … and the statement about closed orbits should be a consequence of calculating the change in angle for moving from maximum to minimum radius.
The procecure felt a bit like so-called early quantum mechanics, where theorems about integer changes in angular momentum were ‘tacked on’ classical theory … and all worked out nicely (and only) with harmonic or 1/r potentials.
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Hm. On reading my copy of Davis’s Classical Mechanics — my old textbook on this — I see he says the kr7 potential allows for closed orbits, but doesn’t say one thing or another about whether every orbit with that potential is closed.
But the section has got that tone like you describe, about early quantum mechanics and other proofs like this, of being ad hoc. Describing where an equilibrium might be is fine. The added talk about what makes it stable? … I suppose that’s more obvious when you’ve got some experience in similar problems, but I remember as a freshman finding it baffling why this should be a calculation. And then the part about apsidal angles, to say whether the orbits are closed, seems to come from a particularly deep field of nowhere.
This does remind me that I’ve got a book I mean to read, partly for education, partly for recreation, that is about introducing the most potent tools of mechanics while studying the simplest orbiting-bodies problems.
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I searched for a reference now – this is the theorem I meant and its proof (translated to English from French): https://arxiv.org/pdf/0704.2396v1.pdf
Quote: “In 1873, Joseph Louis Francois Bertrand (1822-1900) published a short but important paper in which he proved that there are two central fields only for which all bounded orbits are closed, namely, the isotropic harmonic oscillator law and Newton’s universal gravitation law”
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Ooh, thank you. This is interesting. And remarkable for being so compact, too! Who knew there’d be results that interesting with barely five pages of work?
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