Co- authors Wanted for Journal Paper (related to "Jello-O... " thread)

This thread is an experiment to see if publishable math paper can be constructed cooperatively via a forum. PLEASE REFRAIN FROM CLUTTERING IT with "comment post" containing nothing mathematical. If you have already done so, please delete your post.
Please help, if you can. If we show that gravity can not be replaced by any cosmic flux, a paper with the significant helpers as co-authors, will be submitting for journal publication. Certainly, persons who do the integrations are “significant helpers.” Minor contributors will be acknowledge and thanked. The greatest need for help is with 3D integrations, but as the force fields postulated are part of the integrands, they have not yet been well defined. (It is not desirable to make integrands that require numerical integration, if analytic integration is possible.) The following first draft is premature, both because the results are unknown and some critical integrations have not even been attempted. Comments on the clarity of this first draft bellow will be appreciated, but remember that parts are included only to outline the various options, or recruit co-authors, and will be later discarded, when the work is done. Various notes are also sprinkled in the text, which must move to conform to the selected journal’s standards, but for a sciforum post, it seemed more convenient to place notes near their reference in the text.

INTRODUCTION:

Some theories replace conventional mass based gravity with a cosmic flux. It is not uniform near a star with a planet because the planet absorbs some flux, casting a “shadow” towards the star. This lack of local uniformity is how the cosmic flux pushes on the star and planet to supply the net force that allows them to rotate about their common center of mass. It also explains the tidal bulge where the star is expanded into the shadow region. (Rotation, magnetic fields, etc. of star are neglected in this analysis.) Problem is to see if basically spherical stars (sun included) is consistent with gravity due to a cosmic flux. A single significant planet on the z-axis of a Cartesian coordinate system with the star at the origin, is assumed.

A rigorous math answer is sought in this cooperative sciforums effort using the “virtual work” method. - (An initially spherical star is slightly deformed mathematically to an assumed shape to see if it then has lower energy in the force field.) Various “extreme” z-axis-symmetric functional forms of the flux’s force field acting upon the star will be assumed. (“Extreme” in the sense that whatever a particular theory predicts about this flux’s momentum transfer to matter, the predicted force field on the star is likely to have been “bracketed” by at least two of the assumed force fields. This approach is necessary because these “gravity is a cosmic flux” theories seldom are mathematically explicit in their predictions.)

The Center of Mass, CoM, of all shapes used in force and energy calculations is at origin and all shapes have the same volume, V. (Some shapes used only to develop equations do not have their CoM at the origin.) Probably the assumed shapes are neither equilibrium shapes, nor the lowest energy state possible, nor stable. That is, all plasma (and other) instabilities, and any further tendency to change shape are ignored, but all shapes are assumed to be static (no kinetic energy). The drawing below presents the primary shapes considered, but they are only illustrations and not derived from the equations to be given later.

Drawing credits: Needy Bates made the drawing below and QQ's more robust site now holds it. Some changes will be required for publication, but they can wait. The two lables in the boxes need to be reversed. It is the light tan region that is continuous ring around the "belly" of the ellipsoid. - Obvious if you imagine a long skinny ellipsoid, like a pencil just punching thru the sphere. (All are 3D objects, appearing here in 2D.) My intuition tells me that the planet is to the right. I.e. the pointed end of the eggoid is closer to it than the fat end, but like the more important results, we must wait to see what the math tells us.

<img src=http://www.geocities.com/bpstudios/sciforums/Jell-O.PNG>

The following summary gives the current status of our sciforums efforts and indicates specific work (mainly integrations) that remains to be done:

SHAPES . & . EQUATIONS used:

(1) Sphere:
V = (4/3)(pi)(R^3) = (4/3)(pi){(X^2 + Y^2 + Z^ 2)}^(3/2)
Or: (X^2 + Y^2 + Z^ 2) = A = {(3V/(4pi)}^(2/3)
Where “R” is distance of a surface point, (X,Y,Z), from origin.
I.e. radius, R, is the positive square root of (X^2 + Y^2 + Z^ 2).

Note (1): A point in the interior, (x, y, z), of a 3D volume is indicated with lower case letters. Surface points, distance of a surface point from any other point, and constant values (like A above) are indicated by capital letters. (Constants A,B,C,D,E,F,G & V are already used in this post.)

Note (2): R is seldom used because the problem is axially symmetric about the z-axis. More commonly used are the non-italic R or r , which are distances from the z-axis of a surface point and an interior point, respectively.

(2) Ellipsoid (z-axis symmetric):
V = (4/3)(pi){(X/b)^2 + (Y/b)^2 + (Z/a)^2}^(3/2).
Or: (X/b)^2 + (Y/b)^2 + (Z/a)^ 2 = (Z/a)^ 2 + 2(R/b)^2 = A
Which smoothly becomes the above sphere, as “a” & “b” both approach 1. See Note (4) below also.

(3a) Eggoid 1 (z-axis symmetric but “un-shifted” &“un-normalized”):
(Z/a)^2 + 2(Rf/b)^2 = B.
Where “B” is a constant, not necessarily equal to A. See Note (3) also.

The form of Eq.(3a) differs from the final Eq.2 only by a near unity factor or function, “f,” which slightly exceeds 1 for z > 0 but is slightly less than 1 where z < 0 and 1 at z = 0. Eq.(3a) is thus an “eggoid“ (a shape like an egg) because if the final version of Eq.2 and Eq.(3a) have the same values of Z, and B is set equal to A, then the value of R in Eq.2 must equal the value of Rf in Eq.(3a). In this B = A case, where f > 1 the R of Eq.(3a) is smaller than the R of Eq.2. - That is, the eggoid’s more pointed end is in the +z hemisphere, and by the same reasoning, its fatter end is in the -z hemisphere, as illustrated in the right drawing above. Obviously, if f = 1 everywhere, then Eq.(3a) is identical to Eq.2 and the eggoid defined by it has become the ellipsoid of Eq.2, but of a different scale, if B dose not equal A.

Unfortunately, introduction of the factor ”f” has shifted the egg’s CoM into the negative z space, but the CoM can be returned to the point (0,0,0) by a simple translation along the z-axis. See Note 2 for details. This is well defined problem a “co-author” can work on independently.

Two examples of suitable “f” factors are:
……“Linear f”: f(z) = 1 + Dz.
Where here and elsewhere, the “Distortion constant, D,” is a very small positive value. I.e. 0 < D < < < 1.
……“Step f”: f = 1 + D for z > 0 and f = 1 - D for z < 0 and again f = 1 at z = 0.
This discontinuous step function (but a constant inside each hemisphere) is probably more convenient if integrations over the eggoid are done separately for the two hemispheres (+z and -z hemispheres), but all surface tangents that pass thru the z-axis somewhere are discontinuous when crossing the z = 0 plane. This surface derivative discontinuity at Z = 0 should not be a mathematical problem. It certainly is no problem for the “virtual-work” methodology because the assumed shapes need not be physically plausible shape, only mathematically-convenient, continuously-connected volumes that can smoothly transform from the sphere, or back into the sphere, with smooth changes in the parameters of the shape defining equations.

Note (1): Anyone helping to do one of the still undone integrations may chose another form for “f” which is everywhere “near unity” and {f > 1 for z > 0 and f < 1 for z < 0}, but please, if possible, use the distortion constant, D, as your only parameter so that
your eggoid smoothly becomes the above Eq.2 ellipsoid as D approaches 0. It does so (with B = A) for both of the “f” factor functions (step & linear) of the examples above.

Note (2): The adjective “un-shifted” recognizes fact that the CoM of eggoid of Eq.(3a) probably is not at origin, point (0,0,0). This unshifted eggoid is designated “eggoid 1u,” but it needs to be mathematically translated (shifted) along the z-axis by replacing the “Z” of Eq(3a) by the quantity (Z - C). When it has been shifted to place the CoM at (0,0,0) it is designated “eggoid 1s.”
C can be evaluated by integrating the moments of inertia of differential volumes throughout the volume of eggoid 1s with the shift still expressed as “C” and then selecting by inspection the value of C for which the moment of inertia about point (0,0,0) is less than for any other point. (If simple inspection of the total momentum equation does not permit the correct value of C to obtained by inspection of the total momentum expression, then differentiate the total moment of inertia with respect to C; setting the resulting equation equal to zero; and then solving for C.) Thus, C can be expressed in terms of “a,” “b,” B and D only, (assuming no other parameters are used in defining “f”). Consequently, C is not a new parameter, only one convent for writing equations more compactly.

Note (3): The adjective “un-normalized” recognizes fact that the volume of Eq.(3a) eggoids, both 1u and 1s, probably is not V. Before the energy of the eggoid 1s can be evaluated, its volume must be calculated and the equation for it then normalized to have volume V. That is the value of one or more of the parameters a, b, B and D must be changed. B is the obvious and preferred parameter to adjust because changing any of the other three adjusts both the volume and changes the shape of eggoid 1s. The eggoid formed from Eq.(3a), with both CoM at (0,0,0) and the volume normalized to V by suitable choice of B, is designated “eggoid 2.” The eggoid with volume normalized by suitable choice of B, but still unshifted, is called “eggoid 1v.” (Eggoids 1u, 1 s and 1v may useful, because the missing shifting and normalization calculations can be done by one person working independently with the simplier equations, but only Eggoid 2 be used for energy state calculations., unless some other eggoid is constructed with CoM at at (0,0,0) and its total volume is V. (The first of these alternative eggoids to be described here would be called “eggoid 3“, and one is briefly described at the end of this section.)

Note (4):It may be convenient to have both “a” and “b” also defined in terms of the distortion parameter D, so that as D approaches 0, all eggoids and the Eq.2 ellipsoid smoothly transform into a sphere, but not necessarily the sphere of radius R unless B = A. For example we could use:

a = 1 + D and b = 1 - D.
Or: a = 1 + DE and b = 1- DE
Where E is a positive “Eccentricity parameter” whose value partially controls (along with D) the extent to which the eggoid is distorted from a sphere but, despite its value, all eggoids with “a” and “b” so defined smoothly returns to a spherical shape as D approaches 0. Eggoid 2 (and 1v) becomes the original sphere of radius R and volume V as D approaches 0.

Note (5): Anyone helping, feel free to define your “eggoid 4” (Eggoid 3 is immediately belo.) differently if this facilitates an integration, etc. We may need to explore several more eggoids to find one that can be intergrated analytical, or even strange, physically-implausible, non- eggoid shapes. (Recall the text just above Note 1.)

(3b) Eggoid 3 (z-axis symmetric “un-shifted, un-normalized):
{(1/f)(Z/a)}^2 + 2(R/b)^2 = F
From this Eq.(3b), which defines eggoid 3u, eggoids 3s, 3v, & 4, corresponding to those with similar names above, can be defined using the same procedures as above.


FORCE . FIELD . FUNCTIONS:

This section is very incomplete for reasons stated earlier, however, it does seem appropriate to now observe two constraint and give one example:

Constraint (1): The postulated Force Field, FF, must be a continuous, z-axis symmetric, function of (x, y,z). Any differentiable function of “z” alone is both, and probably mathematically very convenient.

Constraint (2): The volume integrals in a sun-like star of the FFs selected for the paper, must be have total force comparable to (or perhaps up to an order of magnitude greater than) Jupiter’s force on the sun.

One FF that could be considered resembles the linear “f“ described above. I.e. FF could be:

FF(z) = F + Gz.
Where F is adjusted to satisfy condition (2) and G is a “Gradient factor” that is related to how far the planet is from the star.

 

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I will get back to you. If Mathematica can't do it then there is no closed-form solution and it will have to be done numerically.

-Dale

 

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I have. For example, pulling can not produce the Taylor instabiliy, that I suggest tears apart all suns held together by push gravity, but will give a new one (really an analogy):

I can pull a wagon with a rope. Can you push one with a rope?

When I mention the Taylor instabily here this is only to specifically name a prior difference that I have in fact several times cited. (I.e. I am showing that your statement that I have never cited any difference is false.) This cited difference may be an error (I do not think so.) but you have never shown it is, so for now the validity of this prior cited difference is in dispute between us, but I have cited it and others.

The pushing on a rope is an example of the Taylor instability. It typically occurs whenever the object being pushed upon can deform to escape the pressure. That is, also illustrated by the case of pushing on Jell-O. etc. Push gravity does to suns what pushing on Jell-O does to Jell-O.

 

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I completely agree. That is why I only suggested a few possible functional expressions for various terms /factors of the integrand and left the specific choice to whoever was going to try to make an analytic integration.

The "virtual work" approach is very powerful and flexible. The shapes used in it need not be the actual distortions that would occur under "push gravity" - See post below replying to MacM which mentions the Taylor instability. It also "feeds" on the fact that the spherical shape is not the lowest energy configurations in most, if not all, cases where only a compressive, pushing, force exists on a non-rigid mass (jell-O, suns, etc).

The shapes chosen for analysis, (their energy compared to the spherical shape) can be very weird if that helps make the integration. All that is required is (1)that they be continuously defromable mathematically to & from the sphere by continuous change of their defining parameters & (2) that they have a lower total energy than the original sphere.

Invent anything you can integrate that satisfies (1)&(2).

If you succeed in showing all push gravities destroy stars, You can be the sole author. I am no longer intersest in publishing anything with possible exception of some ideas I have about mental processes. It may be seem strange, but I lost interest in physics for its on sake more thqn 15 years ago - My concern with it now, and why I wrote a strange physics book is available at the web site under my name. (It is my concern for future of my grandchildern that brought me back to physics.)

 

I don't know what the "virtual work" approach is, but here is the approach I am taking with Mathematica.

First, due to the symmetry involved I chose not to use cartesian coordinates. Instead I used spherical coordinates ({r, theta, phi} http://mathworld.wolfram.com/SphericalCoordinates.html) and prolate spheroidal coordinates ({xi, theta, phi} http://mathworld.wolfram.com/ProlateSpheroidalCoordinates.html) to represent the spherical and ellipsoidal geometries from the post (there is no convenient coordinate system for the eggoid so I didn't attempt it at this time). With both of these systems there is z (north-south) axis symmetry since the pi>phi>-pi coordinate represents the "longitude". The "latitude" is defined by cones making an angle pi>theta>0 with z for the spherical coordinates and by confocal hyperboloids making an asymptotic angle of pi>theta>0 with z for the prolate spheroidal coordinates.

The prolate spheroidal system represents spheroids whose z axis is longer than the other two axes (which are equal to each other). The prolate spheroidal system is characterized by a scale parameter "s" which is half the distance between the two foci of the spheroids. The xi parameter of the prolate spheroidal coordinate system is a shape parameter with the spheroids getting more skinny as xi->0 and more spherical as xi->infinity. So in the prolate spheroidal system I want to define my "star" as having a shape defined by Xi and I want it to have a given volume, V, so I solve for the scale parameter, s, in terms of Xi and V. (Similarly with the sphere where I solve for R in terms of V) To verify that gives me the correct volume I integrated f(xi, theta, phi) = 1 over all theta and phi with xi from 0 to Xi, substituted in the expression for s and got a volume of V as expected. Now I can check my work because I should always get the same answer in both systems in the limit as Xi->infinity.

Now, I was not very sure about the integrand that you wanted me to use. The nearest I could figure was that you wanted f = F + G z. Since the origin of both of these coordinate systems is at the center of the "star" and you wanted your origin at the center of mass of the system I would have to transform it to f = F + G (z - d) where the center of mass of the system is a distance d from the center of the "star".

Look over my approach and tell me if you follow it and agree with it, then be explicit about the integrand you want. I can integrate pretty much any integrand over the sphere/spheroid (wether it has a closed-form solution is a separate question) but you need to be explicit. I will check for your reply after I get back from church.

-Dale

 

I can push a wagon with a rope until you define additional conditions which prohibit such an action.

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So anyway assuming that you really do want to integrate f = F + G (z - d) over the sphere and ellipsoid the results are:

For a sphere of volume V:
V (F - G d)

For an ellipsoid of volume V and with the shape parameter Xi:
V (F - G d)

Basically it is the volume times the mean value of the integrand. Since the integrand is linear in z and the object is symmetric across the x,y plane, the mean is simply the value at z=0. So for the ellipsoid the result of this particular integral is independent of the shape parameter Xi. And obviously it is equal to the spherical case in the limit as Xi goes to infinity.

Let me know what else you would like integrated.

-Dale

 

Briefly, the virtual work method imagines a very-slow* infinitesimal** displacement (or in our case deformation at constant volume) and computes the work this movement does: I.e. the integral (sum) of differential dot products of the local*** force on each differential volume**** element WITH, the local displacement vector integrated over the entire volume - that is the work done. If this net work done is negative, (E.g. moving a weight very slightly higher in any direction has negative dot product.) then the object will not spontaneously move or deform the way assumed. If the work done is positive, then it will move or deform someway, but not necessarily the way you assumed, because your assumed deformation may not be the “path of steepest energy descent.” Some other deformation which is, will occur, but you have shown that the current position or shape is unstable.
___________________________________________________
*Slowly so object never gains kinetic energy. We are evaluating which of two, almost identical**, static energy states/ configurations has the least potential (stored) energy. If all others but the current one and one alternative are prohibited by constraints, the object will go to, or deform to in the case we consider, the “permitted alternative” if and only if, IFF, it is a lower energy state. E.g. Think of holding 1Kg in your had and slowly lowering it vs. slowing raising it. (Note force on it dot product with the “down motion” is positive & with the “up motion” is negative.)
** We consider the integral of “infinitesimal” displacement or deformations because at in no point in the finite deformation/ movement process can work on the object be required if the movement occurs spontaneously. Think of a marble in a saucer on a table. It will not go spontaneously to the lower potential state on the floor as initially it must go up hill to get over the rim of the saucer. If work done as sphere infinitesimally deforms is negative, then we need to consider some “eggoid shape” as this IMHO more likely to be positive and display the instability.
If, however, event the ellipsoid is a lower energy state, I.e. the work done is positive, then we still must show that the deformation continues. Obviously, we cannot do an infinite set of infinitesimals so I think we need to use the “Distortion factor/parameter, D” approach I outline in first post and compare two slightly different D shapes to show that no mater what the absolute value of D is, a slightly greater distortion is also lower energy state. Perhaps we should do this first if not too much more work (the sphere is the D= 0 case etc.)? This seems likely to me if you are planning to use two different coordinate systems and one alone can do the entire task. I would prefer to set up the forces in only one as several of them must also be explored to “bracket” the theory’s force on the high and low sides and yet be “integrateable.“
***In sense of acting on each small differential element of the deformable or movable rigid object.
****Obviously with infinitesimal displacements many parts of homogneious uniform density objects need not be included in the volume integrations because their potential is not changed by the deformation. No need to evaluate these integrals over the entire volume unless we want to consider some inhomogenous internal structure. (and we may someday, but not now.) That is, in the superimposed sphere and “eggoid” drawings of the first post, only the work done in the tan and blue areas, not green area must be considered, if more convenient than integration over the entire volumes. (But of course I am speaking of the corresponding volumes.)

Some old ideas that may help or stimulate your questions/ thoughts:
I was planning to use the axial symmetry to conclude that any force on a volume element not on the z axis would always have a symmetric volume element that cancelled all but the “z component” of the forces on it. Then express only the local z component of my force in the integrand in a one dimensional (dz) integration. I was planning to assume that the differential motion as always normal to the surface of the eggoid as it moves from being a sphere. Thus my dot product’s “cos(&theta; ) was the surface normal unit vector with the unit vector “z” (being careful with the signs.) as the only other factor in that integrand. (multiplying dz, of course, but which I do not call "part of the integrand", do you?)

Your coordinate system may make it even simpler, but I am not so sure because “r“ & “xi“ held constant are surfaces, (very useful if a metal of this shape with constant electric potential is a “boundary condition” in your field equations) but none of your coordinates held constant (I think) are surface normals so, I naively do not see the advantage of non Cartesian coordinates for this “z axially” symmetric problem. BTW I want to confirm that the prolate spheroidal system is just an ellipse rotated about line thru the two focii you mentioned, what I called an “ellipsoid.“

I think we must jointly plan/think how to best attack the two factors of these differential dot products. I think we must assume that the differential motion is always normal to the surface and this is mainly your problem to describe. I think I can construct local force equations of interest (“bracketing” high and low any reasonable theories model) and, mainly, that will be my half of the integrand’s dot products problem.

I will be away for two days, without internet access. Can you tell me a little about what you are thinking about how to express the surface normals part of the dot products? I will work on the force in what ever coordinates you chose but if they remain non Cartesian, I may need more checking with you. Somewhere in a box I have and did bring to Brazil is a very good, hardly read, two volume set of applied math text, (perhaps authors are Feshbach & Morris? - memory fails me?) which even has most of these useful coordinate systems in it illustrated in stereo drawings. (I have no trouble getting each eye to look at a different one - being near sighted makes it easy as I can get close to them and retain good focus.)
Bill

PS by Edit: I only saw you first results after positing (Wrote above off line). After quick skim, I think it is more complicated as you did not know the dot product with surface normals is required. Read above and ask questions. By 4 now.

Post PS - I think we should follow James R's implied suggestion and just ignore all comments posted here that have nothing mathematical to offer. I plan to do so anyway. There are more suitable threads for nonsense. In some sense this tread is an experiment to see if a publishable result can come from cooperative effort of participants in a forum like this one.

 

>> Push gravity does to suns what pushing on Jell-O does to Jell-O. >>

as if

the parameters of a sun push all towards it

 

I think we should follow your (James R's) implied suggestion and just ignore all comments posted here that have nothing mathematical to offer. I plan to do so anyway. There are more suitable threads for nonsense. In some sense this tread is an experiment to see if a publishable result can come from cooperative effort of participants in a forum like this one.

 

Mainly to Dale:
As I was gong to bed, I realized i may be confusing in my longer post below. I speak of the dot product with the local diffferentail element's motion (and that is always correct) but also with the surface normals which is correct only if the main bulk of the sphere and ellipsoid are the same and we are infact integrating over only the volume changed by the change of shapes. That lesser integration is so much in my mind that I sometimes assume that is what we will do and speak as it it were correct in all cases. It is not. Only if the main volumes are homogenious and uniform density and the thus do not change potential as the object distorts. Hope this is clear - I must get up at 5 and it is 1:40 already so will not even edit as I usually do.

 

Billy T as usual. [post=896902]Here[/post]

I found it laughable that he tells people they don't need to understand it (nor does he) to disprove it. None of his proclaimed "Proof" have anything to do with it. He makes false assertions and then tries to claim they prove this or that. He is wacky.

 

Hoping to get DaleSpam's help with math, I said it would not be necessary to go thru all 100 uniKEF thread pages IF it could be regorously demonstrated that ALL push gravity theories are inconsistent with spherical stars (sun).

This thread is an experiment to see if publishable math paper can be constructed cooperatively via a forum. PLEASE REFRAIN FROM CLUTTERING IT with "comment post" containing nothing mathematical. If you have already done so, please delete your post.

This post will be deleted soon, but may periodically appear, if needed.

Perhaps James R will delete his, this, and others that are devoid of mathematical content? I request that.

 

Billy T, this is a forum. If you want to control the content presented, get your own website.

 

Billy, I looked into the virtual work and, at least from what I could pull up on the net, it does not seem to apply. Everything I saw described it as a way to determine the stability of an underdetermined system for solids. In particular it seemed to use a stress/strain relationship to get infinitessimal work. A star is a fluid, so there is no stress/strain relationship. Instead fluids are infinitely deformable. Viscosity is related to shear stress and shear rate, not stress and strain.

I think a potential calculation may be better suited to what you want. I could e.g. evaluate the gravitational potential energy of the two different shapes and show if one is higher than the other. Of course, that would only apply to conservative fields and I am not sure if the push gravities are conservative, but it would certainly work for classical gravity.

As far as surface normal vectors go, that is another reason to use the coordinate systems. E.g. for the spherical coordinates the surface normal vector is simply r (or "r hat" rather) at each point.

Anyway, just give me the integrand you want and I will integrate if possible. I am willing to work in any coordinate system, including cartesian, since now that I have it set up I can translate between them fairly easily.

-Dale

 

MacM, I am not interested in disproving your theory. I have no personal stake in the matter one way or the other (and as I told Billy I am not going to wade through 100 pages of discussion in order to pick a side). I just offered to perform integrations since I have Mathematica and I saw someone mention that there were some difficulties.

I extend that offer to you as well. I would be glad to help you evaluate any integrals that you feel would show your theory as reasonable or valid. I think that you would find that a firm mathematical framework would go a long way towards organizing your theory in your own mind and making it easier to justify to others.

-Dale

 

That is fine, and that is all I was trying to say with "virtual work method," which I agree is usually applied to non deformable bodies. The reason why I tend to think of this stability problem in virtual work terms is that the pushing flux, which is transferring momentum is more directly related to force than potential.

I am not sure how to express the potential of a deformable object in a push gravity field, except by imagining it to slightly deform and calculate the work done. The change in potential and the work done to achieve it (both being energy) are really the same thing. For example:

The work, W, done in lifting a weight is F*D where F is the gravitational force on it and the distant it moves, D, is small enough to consider F a constant. (This is true, even if you imagine it divided into many smaller separate cubes and sum their individual contributions or a liquid and integrate each drop, but to apply this to the liquid or gas or star, the total volume must remain constant.) The change in potential of this weight is Pn -Po = F*D where the subscripts n & o are for new and old positions of the weight.

I will try to supply some more Cartesian coordinates integrands soon, but frankly I am not very optimistic we can show anything but neutral stability for all shapes with f(+z) = f(-z) symmetric & cylindrical about z axis in an isolated star in uniform flux fields. I.e. no support for an ellipsoidal star to be come a sphere or for a spherical star to become an ellipsoid.

I think you have already exhibited this in you first math results. It is intuitively obvious to me, speaking in potential terms (instead of work done) that the thin disk at z = Q, which is dz thick, will have its potential wrt to the plane z=0 elevated by P and the symmetric disk at z = -Q will have its elevated by -P or depressed by P so that there is no net change in potential as sphere becomes ellipsoid or conversely.

This is why I think the "eggoid" which is not symmetric in its distortion for =z and -z exchanged i.e. which is not = {f(z) = f(-z)}] is essential to this project.

Can you help me understand two things:

(1) Make sure I am correct that "conservative" is that the integral around any closed loop is zero. Reason I ask is I thought all simple potential fields had this property. If you are referring to "curl" (&delta; X operator) I do not think MacM even knows what they are, much less has included them in his theory. I recall there is such a thing as the "vector potential", but would need to dig out some texts to understand again. I.e. I too am ignorant now.

(2)What is wrong/ difficult for you/ with me defining (in simple parametric form, as in first post) an Eggoid? I will always make it smoothly degenerate to the sphere by continuous change of the "shape parameter." If I give the complete integrand and the limits of the integration with every thing reduced to a single integration over dz do you think you could attempt it? I think I can do this by first reducing the eggoid to a set of "dz thick disk slices" perpendicular to the z-axis.

There will of course be three different integrals to do. One for the location of the CoM, one for the volume for normalization and one for the actual potential change with slight shape distortion.

 

(1) Yes, conservative means the integral around any closed loop is zero. This works just fine for gravitational potential and voltage (electrical potential), but I don't know if push gravity is conservative.

(2) I should be able to do an eggoid now that I have a little experience here. I just started with the simple geometries, but stopped because I wasn't really sure about the integrand. I do not think you should limit yourself to a single integral over z. Just give me the integrand and the volume you want it integrated over (limits). I can do multiple integrals just fine. I can do vector integrals too.

-Dale

 

I will delete my post once all references to MacM or UniKEF have been deleted. But if you come back with some claim that you have proven UniKEF (or ALL push gravity theories) invalid I will rigorously point out mis-assumptions or other errors where your work does to apply to UniKEF.