Wikipedia:Reference desk/Archives/Mathematics/2019 July 26

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July 26

Name for specific Octagrams?

Are there specific names for the following two *modified* (8,3) Octagrams?

1. ) Connect the corners and the midpoints of a square. (This one is Isotoxal)
2. ) Connect the 1/3 & 2/3 points of each side of a square. (This one is not Isotoxal, but is Isogonal)

Naraht (talk) 20:09, 26 July 2019 (UTC)

Courtesy links: Octagram and Isotoxal figure. Loraof (talk) 23:45, 26 July 2019 (UTC)

And Isogonal figureNaraht (talk) 17:24, 27 July 2019 (UTC)

n-body problem

This is a naive question from a non-mathematician--hopefully an intuitive answer without equations can set me on the right track. The two-body problem has an analytic solution, whereas the n-body problem for n>2 does not. But why can't we solve it this way: we have position-and-velocity vectors B1, ..., Bn whose initial values are known; treat this as a two-body problem for "bodies" B1 and C, where C is the barycenter of B2, ..., Bn. Loraof (talk) 23:37, 26 July 2019 (UTC)

It just doesn't work like that, and I don't remember the details well enough to explain why. But an analytic solution for the general three-body problem would involve solving fifth-degree equations, which generally don't have analytic solutions. Bubba73 ^{You talkin' to me?} 23:44, 26 July 2019 (UTC)

Many body problems generally have chaotic solutions, which can not be expressed in terms of quadratures. Ruslik_Zero 15:05, 27 July 2019 (UTC)

If we can throw some technical math-words at you, the gravitational potential for a three body system (and, by extension, the n-body system) becomes nonlinear even though it is a composition of linear equations; the composition of linear equations that vary in time introduces recurrence in the spatial coordinate of higher order than the initial potential field. The equivalent technical physics-word is that we are treating motion in a non-inertial reference frame.

Conceptually, our chief simplification in a two-body problem relies on rotational invariance - as long as we're "looking" down the imaginary line between the only two objects in our "universe," it doesn't matter how they (and the line) are spinning - because there is nothing outside of these two objects to which we can reference their relative spin. When we add even one extra object - the third body - we can no longer guarantee a single straight line that connects them all - we have broken rotational symmetry, because of a simple consequence of geometry that two points define a line and a third point can't be guaranteed to be colinear - and so we cannot ignore the spinning. As I have said many times before - angular momentum is pretty important to our universe, and it's an even more fundamental property than gravity itself - so if we have it, we can't ignore it. To put it bluntly, this makes the math ... "hard."

If we start with the classical, analytical treatment of the two-body problem, and take advantage of certain symmetries like the ones that lead us to a barycentric coordinate system, we can completely solve the equations of motion. When we add a third body, we immediately see that the symmetry provided by barycentric coordinates is broken: the relative rotations are not identical, which means that at least one of the bodies sees the others rotating relative to each other - in other words, they now have angular momentum that cannot be resolved without the use of a rotating coordinate frame.

In Marion and Thornton's textbook on classical analysis of dynamical systems, chapter 10, they work this problem and demonstrate how it immediately yields the coriolis force and centrifugal force; and the tidal force; and the equations that govern these forces are - while still analytical - described by large Lyapunov exponents. This is the mathematical way that we say "the equations are too hard." In Chapter 4, the authors describe the very subtle distinction between complex and random; the distinction between deterministic and predictable; and the distinction between the overlapping concept of non-linear and chaotic equations. To quote their introductory text, "researchers in many disciplines have come to realize that knowing the laws of nature is not enough. Much of nature seems to be chaotic."

So - while our OP has proposed a simplification - say, using an R-tree to subdivide the universe into spatial sub-regions, and computing net potentials between pair-wise very-distant "groups" of two-body approximations - it turns out that this is just a really bad, really numerically unstable way to compute the problem. It turns out that other, more non-obvious mathematical techniques are better, often solving (or at least estimating) by converting into a transform-domain or other coordinate generalization.

Nimur (talk) 18:05, 27 July 2019 (UTC)

The short answer is that the total gravitational force exerted on $B_{1}$ by $B_{2},B_{3}$ is not equal to what you would calculate assuming that all the mass of $B_{2},B_{3}$ is concentrated at their barycenter. Thus, calculating as you suggest will lead to incorrect results.

If $B_{2},B_{3}$ are close to each other, approximating them by their barycenter is more accurate, and the model is useful. That's why you can reasonably use it to model the system of sun, Earth, moon - the Earth and moon are very close, so replace them with their barycenter for the purpose of orbiting the sun, and then superpose the moon's orbit around Earth. (It also helps that the mass of the moon is quite small.) -- Meni Rosenfeld (talk) 00:19, 28 July 2019 (UTC)

Consider a three body system: Earth, Moon and yourself. Do you experience a gravitational force towards the Earth-Moon system barycenter, or rather to the Earth's surface? --CiaPan (talk) 08:23, 28 July 2019 (UTC)
Forgot to ping: @Loraof: --CiaPan (talk) 08:25, 28 July 2019 (UTC)

The barycentre of the Earth-Moon system is beneath the surface of the Earth. 2A00:23C4:7916:5100:BCA8:3981:9551:8F54 (talk) 10:01, 28 July 2019 (UTC)

Good, very well spotted. So now, what about an astronaut, standing on the Moon's surface...? (at the side facing the Earth, of course!) --CiaPan (talk) 11:04, 28 July 2019 (UTC)

CiaPan, I'm not sure what you're getting at. Both bodies exert a gravitational pull, and there is a net gravitational pull that we can compute by simple summing of the vectors. The direction of that net force vector depends on where the test particle is - in your example, the test-particle is the astronaut. There is no specific physical reason why the surface of the planet or moon needs to coincide with any specific value or direction of the net gravitational pull. In fact, there is a specific physical reason that precludes that surface from coinciding with the "changeover point" where the net force vector changes direction - and that imaginary surface defines what we call the Roche limit, a theoretical value that is computed during the very same mathematical tidal-force analysis described above.

Surely you aren't suggesting that the net force of gravity should pull the astronaut off of the surface? If that were the case, ... it would also imply that the surface of the lithosphere would not be geologically or cosmologically stable, and it almost-definitionally would not have formed at that particular spot, as measured in relation to the pertinent n bodies that produce the net gravitational potential.

I think you are conflating the near field and the far field. Our article on this topic only describes it in electromagnetic terms, but it is generalizable to any potential field (including gravity). Another useful article is our multipole expansion topic, which presents us with additional options for approximating a complex potential that models the net gravitation of n bodies. This does not, however, make the difficult math go way - the field is still time-varying, non-inertial, and so on - but at least these tools of analysis give us methods to quantify and constrain our physical approximation methods.

Nimur (talk) 17:41, 28 July 2019 (UTC)

CiaPan is illustrating Meni Rosenfeld’s correct answer. (If only other contributors were similarly terse and on-point, so that the correct answer wasn’t buried under so much dross....) JBL (talk) 11:50, 29 July 2019 (UTC)

Right, thanks. I dare say that while there's a lot to learn from Nimur's answer, it completely misses the mark on the OP's question. The OP asked why a specific method to solve the problem doesn't work, and the answer to that is quite simple. I've pointed out the key observation, and CiaPan illustrated it, by considering that the force exerted on an astronaut by Earth+Moon is not in the direction of the Earth+Moon barycenter - contrary to the OP's assumption that you can replace a collection of bodies with a point mass in their barycenter. -- Meni Rosenfeld (talk) 13:28, 29 July 2019 (UTC)

Nimur I apologise for a delay - I'm on vacation now, with quite limited access to Wikipedia. My point with the Earth-Moon example is that the actual gravitational force may well be opposite (or generally: unrelated) to a direction to the centre of mass of the playing bodies. Hence reducing them to a point mass at their mass center is often counterproductive. One needs strong reason to do such simplification. --CiaPan (talk) 17:05, 31 July 2019 (UTC)

Newton showed that the gravity field of a body with spherical symmetry is equivalent (outside itself) to that of a point mass. That's not true of an arbitrary collection of masses. —Tamfang (talk) 02:28, 31 July 2019 (UTC)