A Story of Inertial Interactions and Cascades

[kavaris]

Preface
This might be hard to grok at first since it is combining physics/science or engineering in a storytelling/artistic or mystical—as well as in a self reflective and figurative way — ALAS, it is how i chose to  blend  these things together, in such a way as to make this problem more accessible and relatable to others — And from a perspective that makes three body sorts-of problems (as well as my own equilibrium snapshot version of...) a little clearer and relatable to normal physics, and areas o' math, and all these different things that i had said previously...

Intro
So by now, yous most likely may be familiar w/ the three body problem. Yous might also have seen or heard of what a juggler/entertainer is, who juggles three balls or bottles? I* only ask so that when i say *cascade*, that you understand what motion im referring to (as opposed to the "cascade" used in a general sense)
This of course expands into lots of different fields and circular trajectories and different types of hypothetical objects in motion(s), however i only mean to start from a classical, physical description of,.. recognizing the "ideal scenario", so that, yous too may be able to have some idea of what would happen, and continue to try understanding it. Let us begin by assuming that these  bodies  or  objects  are of  relatively  the same shape, size, and traveling at a vaguely similar sort of velocity. Yous may read this as a story, as an instruction/guide, or as a formula, it doesnt matter in that regard...

Part 1: C Approaching A and B

In the beginning, A and B are in orbit around each other (this is a kind of simple prerequisite, that which isnt necessarily important unless we need more accurate numbers—math and/or  physics equations for):  A and B are gravitationally interacting, and we’ll assume their orbital relationship is stable for now. They create a system that could resemble a binary orbit or a close, synchronized orbital configuration. Their masses and velocities are similar, so neither dominates the other.

Now C approaches from OUTSIDE the orbit of A and B: Body C approaches the system from a larger orbit (around A and B), perhaps from a more elliptical trajectory. This is crucial because if C were to approach directly from the center or in a perpendicular motion, it could disrupt the balance and either get pulled in too strongly or leave the system entirely. Instead, C needs to approach tangentially in such a way that it doesn't immediately fall into an orbit of just one body but interacts with both A and B.

C’s path influences A and B: As C approaches A and B, its gravitational influence begins to affect their orbits. This could cause small perturbations, but these adjustments aren't chaotic yet. Rather, they slowly tweak the orbits of A and B, drawing them into a new interaction with C at the outer edges of their system.

Part 2: Crossing the Center

Once C is close enough and has been interacting with the gravitational influence of A and B for a while, it will reach the point where it crosses through the center of their orbit. This moment is key to the system's temporary stability and the formation of the figure-eight orbit:

C crosses the center of the system: This marks the first real interaction where all three bodies are in a sort of direct gravitational feedback loop. The crossing creates an exchange of momentum and gravitational pull that "pulls" C through the center of A and B's orbit. In turn, the forces exerted by A and B on C change the trajectories of all three bodies, temporarily aligning them in such a way that they seem to follow a figure-eight pattern.

To be more accurate, i dont know exactly how this plays out, i only imagine that it could eventually play out wherein each begin rotating around one another, however brief it could be...

As C crosses through the center of A and B's orbits, it will also influence their velocities and positions. A and B might now adjust their own orbits to compensate for C's mass and momentum, continuing to orbit each other, but with an additional gravitational pull from C that creates the harmonic balance.

Eventually

This exchange of forces causes a temporary equilibrium where the bodies seem to perpetually "cascade" around one another, following paths that interweave and form the figure-eight shape, that is until those slight perturbations (or changes in velocity) destabilize the system, as over time, even small variations in velocity or position lead to a loss of this perfect harmony. As the bodies continue to influence each other, the orbits might slowly become more elliptical, or rather, the bodies shift into a more chaotic state. The figure-eight pattern will break down as the forces grow too complex and unpredictable, which results in irregular orbits and erratic, chaotic paths (they do of course collide too) But the "Thee Body Problem", if im correct, specifically means to name that difficulty of predicting their motion (from any such kind of scenario) given the non-linear nature of their interactions.

p.s. anytime ive done psychedelics, i have always seen this very poignant motion or shape, portrayed as equi-distant circles in orbit with one another, and so i thought that it would make for an interesting narrative if described in the context of planetary orbit. It also got me looking into the overlap between rules that involve cascades, and rules that involve multiplicative properties and such... and the two most interesting/relevant areas that came up were called Exponential Growth / Exponential Decay (Multiplicative Cascade) and Coupled Oscillator Systems (Resonance and Cascade), both used in physics and electrical engineering.

If you know about this topic, or anything thats tangentially related, or just anything you happened to think of while reading, feel free to post it here, as that is the reason that i created it, such that it can get your juices flowing, and brains working today or whenever youve come upon it.

Also if you are someone who wants to try simulating this on a computer or something, and want to understand the exact initial conditions that lead to the *greatest resonance* and *longest equilibrium between*, id be happy to hear about it, or to even be apart of the experience w/ you.

Edited 20 hours ago by kavaris

[LastThursday]

Interesting. I think the Three Body Problem simply put is that there is no closed form solution to the equations, so no final equation you can plug numbers into to predict the motion. You can predict the motion with numerical methods (i.e. an algorithmic simulation), but this could diverge from reality over time due to inaccuracies in the initial numbers and resolution of the simulation.

An external body C's trajectory also may not pass through the space between the bodies A and B, but some way off to the side. Two bodies will revolve around a barycentre, which is a centre of mass of the A and B system. So it may not have a figure eight configuration if this happens.

Exponential growth and decay (damping), is closely linked to oscillation, in that they're both instances of solutions to differential equations: essentially equations that feedback into themselves. This comes from the infinite series expansion of the exponential function, from which can derive both the exponential function and trigonometric functions. Basically any oscillation is the tug of war between two opposing exponential actions/forces. Circles and ellipses arise out of the motions because these are oscillatory motions, the opposing forces being inertia and gravity.

The case of gravity is interesting, because bodies must somehow "communicate" their presence to each other. Since the force between two bodies is dependent on their distance to each other, this distance must somehow be communicated between bodies (i.e. force or warping of spacetime). But no communication can happen instantly, and so there is always a lag dependent on distance. The upshot is that the force felt between two bodies is not the instantaneous force, but a delayed one. For example if the Sun suddenly disappeared, it would take 8 minutes for that gravitational effect on Earth to "update". Ultimately, this is the cause of gravitational waves and frame dragging. If you want to simulate things correctly, this effect needs to be taken into account. Gravity can also not be shielded from (that we know of), so it has infinite reach - everything tugs on everything else in the universe. So there are no instances of an isolated system, and any simulation can only ever be an approximation, even if you had infinite precision.

I'd say there isn't much difference between a two body problem and three body problem. Both will suffer from numerical precision problems. You can't know initial conditions with complete accuracy and you can't compute most functions with complete accuracy either, even with a closed form solution. The exponential function is an infinite series of computations. But there will be solutions to the three body problem which are closed form and so form stable orbits.

Edited 37 minutes ago by LastThursday