July 2015 - mathfun@mailman.xmission.com

[math-fun] Lienard–Wiechert potentials
by Warren D Smith 19 Jul '15

19 Jul '15

The wikipedia article I cited also gives the equations for the fields. Take beta=0 in those equations (stationary observer) and you will find the fields arise from retarded not instantaneous positions of source charges, and this also happens if the source charges move at constant velocities.. Perhaps the source of ES's confusion is he is thinking of situation where all charges (source and observer) move at equal velocity, i.e. all are stationary in my ref frame.

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[math-fun] Retarded Maxwell charges and approximate generale relativity
by Warren D Smith 19 Jul '15

19 Jul '15

Lienard–Wiechert potential: https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential and https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential#Defin… (Simplest when beta=0) the retarded potential is generated by where the source charge was a speed of light delay ago, and the corresponding fields ditto. (A statement by Salamin had seemed confused re this.) Newton law with retardation added is not the correct equations of gravity but hopefully are less-incorrect than plain Newton law. Retarded Newton law as I pointed out earlier would cause weird effects like non-conservation of angular momentum in symmetrical rotating scenarios. The post-whoever equations of approximate general relativity all involve retarded locations, but they also involve (what a Newtonian would call) multi-body forces. These equations recast GR in a Newton-like framework and they have been worked out at Kth order for K=1,2,3,4,5 at least, I'm not sure how far they've gone. Last time I checked I think it was not known whether at least one of these sequences can be continued forever, or whether some obstacle arises at some high order. There are two such sequences. One is based on a power series expansion about c=infinity. The other is based on G=0. The first approximant beyond plain Newton is used by NASA to simulate the solar system.

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[math-fun] H Baker on gravitation funnyness in galaxies
by Warren D Smith 19 Jul '15

19 Jul '15

> HB: > Some of the evidence for "dark matter" ... > We know that there is a supermassive black hole > consisting of 4.1-4.5 million solar masses at > the center of the Milky Way: > https://en.wikipedia.org/wiki/Milky_Way --supermassive BHs in galaxies are only, like, 0.5% of the galaxy mass, or less. So they don't really matter for this, but they do matter in the sense that they seem to stabilize galaxies, and may even be crucial for that. > Our Sun is roughly in the plane of the Milky > Way, and "sees" forces from every bit of mass > in the Milky Way. I claim that our Sun "sees" > larger-than-expected forces from masses approx > 180 degrees around the Milky Way due to > *gravitational lensing*. --this effect should be negligibly small. The optical lensing should be of the same order, and it clearly is negligibly small. > I'd love to know if this gravitational > lensing effect is very strong compared > with the usual centripetal forces on the > Sun. In particular, what % addition > speed does this lensing effect add to > the speed of the Sun around the center > of the Milky Way? --It'll be a tiny amount. Well below 1%. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] H Baker on gravitation funnyness in galaxies
by Henry Baker 19 Jul '15

19 Jul '15

Gene: The basis of this caution is the fact from electromagnetic theory that the electric field of a charge moving at constant velocity is directed towards (or away from) the instantaneous position of the charge. It does not point to where the charge was at an earlier time as if the field propagated at the speed of light. Warren: --supermassive BHs in galaxies are only, like, 0.5% of the galaxy mass, or less. So they don't really matter for this, but they do matter in the sense that they seem to stabilize galaxies, and may even be crucial for that. ------------ Points taken. Re Gene: I was going to suggest using the GR ray-tracing program developed for the movie "Interstellar", but if they've taken Gene's comments to heart, then perhaps the Interstellar SW might not do such a good job calculating the gravitational lensing for focussing *gravitational forces*. It would be interesting to ask the Interstellar folks in any case. In any case, Gene's comments might change exactly where a focal point might be and what the point spread function would look like, but it shouldn't change the fact that there *is* a focal point. Re Warren: Of course, the galaxy lensing effect isn't just due to the black hole itself, but to the entire amount of mass influencing the curvature of the space that the gravitational force travels through; the black hole is one certain place where the space is guaranteed warped enough to cause a lensing effect. Furthermore, if there really is "dark matter", then that should cause even greater lensing effects. I continue to believe that any black hole in the galactic center would be the "same" black hole for both normal matter and for dark matter. Perhaps that assumption is wrong?

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[math-fun] Donald Saari on Dark Matter
by Warren D Smith 19 Jul '15

19 Jul '15

Clark & Meeker sent me Saari's paper. One thing that is ludicrous in his paper is stability. Saari says he will consider a solution of the equations of motion to be "stable" if it keeps the moment of inertia time-invariant. That is like saying that a pencil balanced on its point is "stable" because its potential energy, according to the exact solution, stays constant. Utter hogwash, and Saari sacrifices most of any teeny tiny hope he might have to producing physically relevant results and being taken seriously, with that decision. What is the actual stability status of Saari's "spider web" configurations? You will not find out by reading his paper.

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[math-fun] dark matter evidence; gravitational lensing
by Henry Baker 18 Jul '15

18 Jul '15

Some of the evidence for "dark matter" involves the excessive force holding stars further out on the spiral arms of a galaxy, so that they can revolve faster about the center. We know that there is a supermassive black hole consisting of 4.1-4.5 million solar masses at the center of the Milky Way: https://en.wikipedia.org/wiki/Milky_Way Our Sun is roughly in the plane of the Milky Way, and "sees" forces from every bit of mass in the Milky Way. I claim that our Sun "sees" larger-than-expected forces from masses approx 180 degrees around the Milky Way due to *gravitational lensing*. This gravitational lensing will affect the light received from the points directly opposite, but due to the masking effect of the center of the galaxy, the lensing effect on light is smaller than the lensing effect on gravity, because there is no masking effect on the gravity force from the opposite side of the galaxy. This gravitational lensing effect also operates out of the plane of the galaxy, so points 180 degrees opposite the Sun are *focused* on the Sun. One could presumably compute a "point spread function" (PSF) of this gravitational lens in a manner similar to the PSF of any other lens. This "cross-galaxy" PSF will depend upon the radius from the center of the galaxy, so there are almost certainly values of the radius for which this PSF is especially well focused. I don't know enough about GR to actually calculate this PSF, but I would imagine that this calculation isn't particularly difficult to do analytically, but even if it requires numerical simulation, it still shouldn't be a particularly challenging calculation. I'd love to know if this gravitational lensing effect is very strong compared with the usual centripetal forces on the Sun. In particular, what % addition speed does this lensing effect add to the speed of the Sun around the center of the Milky Way?

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[math-fun] Saari spiderweb exact solutions of Newton laws
by Warren D Smith 18 Jul '15

18 Jul '15

It seems to me, Saari's "spiderweb" exact solutions may be mathematically interesting. This is in spite of any complaints I have re Saari's hype about dark matter, and his enormous misuse of the word "stability." Why are they interesting? First of all, if you have some matter, and it is subject only to Newton's laws, and it has total momentum=0 and nonzero total angular momentum=A, then if you make the matter lose energy subject to those 2 constraints, it will eventually reach a state where all the matter lies in one plane (orthogonal to the A vector). So "discs" shaped much like many galaxies, and "saturn ring" configurations, thus arise naturally. I think the importance of discs was first realized by P.S.Laplace (?). The Saari spiderwebs are exact solutions, and they do lie in a plane. So in that sense they seem realistic. Second, the Saari spiderweb rotates as a rigid body. This is completely unlike actual galaxies and saturnian rings. Actual disc galaxies rotate with fairly "flat velocity curves," meaning the velocity v of the rotating stuff (which lies in the plane and orthogonal to the radial vector) stays approximately constant as its radial distance to the center changes. This empirical fact, first highly publicized by astronomers Vera Rubin and W.Kent Ford in the 1970s, is attributed to a conjectural halo of invisible "dark matter." A rigid body, on the other hand, would have linear velocity curve v=const*r. And a uniform matter density in the plane, would have a velocity curve v(r) obeying const*r^2 / r^2 = v^2 / r i.e. a "square root curve" v = const*r^(1/2). [If the matter in the plane were to have an areal density(r) correct to cause a flat velocity curve then that density presumably would need to obey const/r^2 * integral(0 to r) density(r)*r*dr = v^2 / r i.e. the density would be density = const/r.] And finally, the Saturn rings should have velocity curve obeying v^2 / r = const / r^2 i.e. v = const*r^(-1/2) the inverse-square-root curve. So the collection of powers of r we have here is {-1/2, 0 1/2, 1} for velocity curves v=const*r^P. Henry Baker has asked: "what would Saari's exact solutions do in general relativity rather than Newton?" Or about GR and galaxies. Normally GR is considered pretty irrelevant to galaxies because almost all motions are way slower than lightspeed, and the Einstein cosmical constant is too small to affect things on galactic length scales, so Newton works quite well. I would presume that exact solutions of GR much like Saari spiderweb, should exist, but nobody ever found them and they are probably not expressible in close form. However, there are a sequence of better and better approximations to GR, called the "post-Newtonian" and "post-Minkowskian" approximations. The first is Newton's laws. I believe the second is used by NASA to simulate the solar system. It seems to me that within any one of those approximations, exact Saari spiderweb solutions should exist, and they should exhibit essentially the same degree of difficulty to find them numerically, as it is to find Saari's Newton-law spiderweb exact solutions numerically. Another interesting remark is this. Among all spiderweb solutions, some small subset of them ought to be distinguished as "optimum" in some respect. What kinds of optimality would you want, and what spiderwebs would result? Finally, there are some unusual galaxies out there. Take a look at "Hoag's object" https://en.wikipedia.org/wiki/Hoag%27s_Object ---- PS. The "bullet cluster" although it has been hyped as a "proof" of the existence of dark matter, can also be (and has been) argued as proving the quantitative failure of the dark matter model. So be careful, there is a lot of hype about dark matter supported by very shallow and poor analyses.

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Re: [math-fun] ArcCosine Miracles List, and explanation
by Bill Gosper 18 Jul '15

18 Jul '15

On Sat, Jul 18, 2015 at 4:01 AM, David Jeffrey <djeffrey(a)uwo.ca> wrote: > Derive (God bless its bits and bytes) can simplify all of these > expressions automatically. > It does it by reducing each acos to a canonical atan. Then the two atans > cancel. > > Derive, you may remember, was bought by TI and then withdrawn from the > market. > > David Jeffrey > U. Western Ontario You bet I remember. Albert Rich sold it to them but they paid him to continue enhancing it. He hired me to help. We made exhilarating progress for eight weeks. Then TI pulled the plug. They seem to be a completely uncreative, green-eyeshade company, but I'm not sure they bought Derive with the intention of destroying it. Danny Hillis has some amusing stories of their corporate espionage endeavors. There was also some craziness about them de-implementing the Gamma function on their calculators because it gave students in certain countries an unfair advantage. They also knocked off the Symbolics Lisp Machine(s) (TI-Explorer). But it's unclear they had much to do with the death of Symbolics, as much as they might have welcomed it. --rwg > > > On 2015-07-16 18:56, rwg wrote: > >> On 2015-07-16 08:27, Warren D Smith wrote: >> >>> It is interesting that some of the most-amazing among the listed miracles >>> involve only powers of 2 in denominators: >>> >>> 5*acos(-7/8)+2*acos(-61/64) =1080.0 >>> 4*acos(-57/64)+5*acos(-31/32) =1440.0 >>> 1*acos(57/64)+5*acos(-3/4) = 720.0 >>> 1*acos(61/64)+5*acos(-1/4) = 540.0 >>> 4*acos(-3/4)+1*acos(-31/32) = 720.0 >>> 2*acos(-3/8)+1*acos(-23/32) = 360.0 >>> 2*acos(5/8)+1*acos(7/32) = 180.0 >>> 3*acos(-17/32)+4*acos(-11/16) = 900.0 >>> ... >>> >> >> Hey, 3-4-5 ! >> In[905]:= 3*ArcCos@(-17/32) + 4*ArcCos@(-11/16) == 5*Pi >> >> In[906]:= FullSimplify[%] >> >> Out[906]= 3 \[Pi] == 6 ArcSin[17/32] + 8 ArcSin[11/16] >> >> is dumfounded, too. >> --rwg >> >> > -- > You received this message because you are subscribed to the Google Groups > "mathfuneavesdroppers" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to mathfuneavesdroppers+unsubscribe(a)googlegroups.com. > For more options, visit https://groups.google.com/d/optout. >

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[math-fun] Square-wheeled trike essay
by James Propp 18 Jul '15

18 Jul '15

The essay is now live, at https://mathenchant.wordpress.com/2015/07/15/the-lessons-of-a-square-wheele… . (I removed the calculations that initially gave me trouble; I eventually got them to come out right, but I couldn't muster much enthusiasm for them, so I figured my readers wouldn't either.) Comments are welcome, especially those that I might be able to apply in future columns. The intended audience is the sort of people who read Martin Gardner's column. Jim Propp

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[math-fun] Rockets and arccos identities
by Warren D Smith 18 Jul '15

18 Jul '15

The USA did indeed beat the Russians in creating the first artificial satellite, they exploded a rocket nose containing a lot of ball bearings and explosive. Result, some of the balls went into orbit. This was well known and was announced. It was, however, an extremely stupid idea for the future of space-trash. It is impressive how awesomely irresponsible a lot of power-mad US scientists and/or those employing them, were, another example being the mega bombing of, and leaving of radioactive waste on, various Pacific atolls for absolutely no good reason. The story about the manhole cover being the first thing humans made escape the earth is amusing, but could it really survive passage thru the atmosphere and exposure to the radiation from the nuke? ----- All the arccos identities (as I explained in my first post) can be proven using complex number arithmetic involving integers and sqrt(integers) only, no trig or arc-trig needed. Schroeppel then posted one explicit example of such a proof. It is trivial to get mathematica to prove them in this way, although it almost always is not smart enough to prove these identities if they are given in the form I gave them involving acos (or maybe it is capable of that, but if so you'd have to know more about mathematica than I do; naively using their usual simplifiers cannot do it). I was aware of the Machin-like arctan identities, although I do not understand the latest greatest technology by Arndt, Nimbran, et al for finding those of very high order. Unlike the arccos identities, those can be proven using complex arithmetic involving integers only, no sqrt(integer) needed. Also of course the arccos identities can be re-expressed using arccos(a/b)=arcsin(sqrt(1-(a/b)^2)) if desired, or arccos(a/b)=arctan(sqrt(b^2-a^2)/a). For this reason I suspect my arccos identities are a wider class than the Machin-like arctan identities. The reason I was interested in them and the rational-rational triangles ( https://dl.dropboxusercontent.com/u/3507527/RatRatTriangles.txt ) was because they arose inside gear-configurations by William Somsky. In some cases the arccos identities can be proven without need of any sqrt(integer), such as arccos(a/b) when a^2+c^2=b^2 arises from pythagorean triple. However such all-integer-proofs usually are not applicable. I conjecturally know all rational-side-ratio, rational-angled triangles in all 3 geometries, but currently only am able to prove the conjecture in the Euclidean geometry. For some reason I do not currently understand, the arccos identities like to employ powers of 2, 3 or 5 as denominators: 1*acos(-31/81)+2*acos(-5/9) = 360.00000000000 1*acos(-17/81)+4*acos(-1/3) = 540.00000000000 1*acos(17/81)+2*acos(-7/9) = 360.00000000000 1*acos(47/81)+2*acos(-8/9) = 360.00000000000 1*acos(49/81)+2*acos(4/9) = 180.00000000000 1*acos(73/81)+2*acos(2/9) = 180.00000000000 1*acos(79/81)+4*acos(-2/3) = 540.00000000000 1*acos(-47/128)+2*acos(-9/16) = 360.00000000000 1*acos(-7/128)+6*acos(-1/4) = 720.00000000000 1*acos(7/128)+3*acos(-7/8) = 540.00000000000 1*acos(41/128)+2*acos(-13/16) = 360.00000000000 1*acos(47/128)+6*acos(-3/4) = 900.00000000000 1*acos(79/128)+2*acos(7/16) = 180.00000000000 1*acos(97/128)+2*acos(-15/16) = 360.00000000000 1*acos(103/128)+2*acos(5/16) = 180.00000000000 1*acos(115/128)+3*acos(5/8) = 180.00000000000 1*acos(117/128)+3*acos(-3/8) = 360.00000000000 1*acos(119/128)+2*acos(3/16) = 180.00000000000 1*acos(127/128)+2*acos(1/16) = 180.00000000000 4*acos(-117/125)+6*acos(-24/25) =1620.00000000000 2*acos(-71/125)+3*acos(-23/25) = 720.00000000000 1*acos(-44/125)+3*acos(-4/5) = 540.00000000000 2*acos(44/125)+6*acos(-3/5) = 900.00000000000 1*acos(71/125)+3*acos(-1/5) = 360.00000000000 2*acos(117/125)+6*acos(-4/5) = 900.00000000000 1*acos(118/125)+3*acos(-2/5) = 360.00000000000 I currently have no idea why that is, but it would be nice to understand it, e.g. recognize pattern, produce closed forms. I'll remark that in some ways, the "acos miracle" identities are analogous to Fermat's last theorem (Wiles) and the Catalan conjecture (proven by Preda Mihailescu). Except that those were about nonexistence, while the acos miracles do exist, plentifully. A table of AcosMiracle identities (as well as some "near miss" inexact examples I computed mainly to get an idea of how much precision was needed) along with some theory, is now posted here: https://dl.dropboxusercontent.com/u/3507527/AcosMiracles128.out One of the most impressively close near misses is 3*acos(89/119)+99*acos(38/53)-25*pi=4.333...*10^(-13).

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