SpaceMatter has attributes. The basic ones which I may take as primitives are fractal trochoidal motions at all scales, fractal density at all a scales , fractal viscosity at all scales.
I can detail this arbitrary motion at all scales as net pressure driven vorticity both accelerative and constant at all scales. These next 2 videos give an introduction to Kelvin and Helmholtz vorticity kinematics.
Let's take a look at the curl.
This is a determinant, and as such it calculates an area and volume. However, instead of summing the resultants , they are attached to unit directions making the difference of the differences into a vector. The pointing of this vector is therefore dependent on how different the changes are between the velocities in the 3 directions.. This contra cylic mixing of the velocities compared by subtraction reflects and magnifies differences as one moves cyclically around a volume. Spherical volumes that differ by radial expansion, or contraction come up zero. Thus this gives a measure of how far from spherical expansion and contraction a space deformation is.
The particular differences that give a non zero result involve centripetal and centrifugal differences. That is to say expansion with contraction occurs in deforming the spherical volume. This is the case in the strain ellipsoid.
One has to be careful when interpreting the curl. Conventions of 3 directions are misleading. 6 basis vectors are involved in 3 contra pairs to consequently span the volume. The coefficient relationships must define a centripetal centrifugal oscillatory relationship to give a non zero result. However the arbitrary relationships are trochoids or roullettes not necessarily ful encirclings of the reference frame origin. Local circular motions are thus possible. I study Laz Plath's videos to keep this fresh in mind.http://youtu.be/rz8A5l_yn34
i had to googlr chenrich trochoids to find this page. Enjoy and explore.
I'm really giving my numbers a workout. They represent magnitudes, but also positions in the plane.
We can find an expression for the coordinates of an arbitrary point on a given trochoid. It is convenient to represent points in the plane by complex numbers. (Here is a refresher on complex numbers.) Let the supporting circle have radius 1, and center 0. Initially, let the roller be tangent to the supporting circle at 1; the center of the roller is at C0 = (1-w). Let the vector from C0 to the initial position T 0 of the tracing point be ρw. Let the arm C0T0 (extended if necessary) intersect the roller at V0 = P0.
Now let us move the roller so that its point of contact with the supporting circle is P = eiθ = cosθ + i sinθ. Then the center of the roller moves to C = (1-w)eiθ. Let T be the position to which the tracing point moves, and let the intersection of the arm with the roller move to V. The arc length from C0 to C is θ; therefore the arc from V to P is also θ, so the angle from CV to CP must be (1/w)θ. Now θ specifies the direction of CP; so the direction of CV must be specified by (1 – 1/w)θ. Therefore the vector from C to T is
We have found
T = (1-w)eiθ + ρwei(1-1/w)θ .
This is a "parametric representation" of the points on the trochoid. We will say that θ is the parameter of the point T. (For this particular trochoid, and this method of generating it.)
It is convenient to denote the value of T by T(w, ρ)(θ). Then T(w,ρ) is a function from real numbers θ to complex numbers denoting points in the plane. We will also start using "T(w, ρ)" as a name for a trochoid with wheel ratio w and arm ratio rho;. When ρ = 1 we will omit it; this "T(w)" is the name of a cycloid.
It is natural to multiply the expression T(w, ρ) by a constant, to change its scale or rotate it; and to add a constant, to shift its center from the origin.
Double Generation of a Trochoid
This formula for T shows that a trochoid is generated by a combination of two circular motions. The first is associated with the motion of the roller around the stationary center; it has period 2π and amplitude (1-w). The second is associated with the rotation of the roller around its own center; it has period 2πw/(w-1) and amplitude ρw.
Or, we could define Q = ρei(1-1/w)θ and note that
T = (1 – w)P + w Q .
Thus P and Q move around two circles of different radii, at different rates; and T divides the segment PQ in the constant ratio w : (1-w).
This suggests that we could think of Q as the contact point of a rolling circle on which is carried the vector going from wQ to T. The result is a second way of generating the same trochoid. The fact that each trochoid can be generated in two ways was discovered by Daniel Bernoulli in the seventeenth century. Here is how the two ways of specifying the trochoid are related:
Double Generation Theorem: If a trochoid is generated with a supporting circle of radius R, wheel ratio w, and arm ratio ρ, then the same trochoid can also be generated with a supporting circle of radius R′, wheel ratio w′, and arm ratio ρ′, where
R′ = ρR,
w′ = 1 – w, and
ρ′ = 1/ρ .
In our algebraic notation, the trochoid R·T(w, ρ) is equivalent to Rρ ·T(1-w, 1/ρ).
As I said above, a trochoid is called "prolate" if ρ > 1 and "curtate" if ρ < 1. The double generation theorem seems to me to make this distinction somewhat unreal.
Here is an applet which demonstrates double generation. The controls are like those of the applet farther up on this page; their values refer to the blue circles and arms. The values of R′, D′, N′, and ρ′ refer to the green circles and arms.
I have to say, that I am still amazed at how these invisible vorticity flows are so clearly revealed in these simple ways in fluid mechanics. In particular, the Twistorque and Twistorque vectors and relationships I struggled so hard to justify are immediately apparent in fluid mechanics. The use of Newtonian principles is well justified at this level, and reflects his thoroughness in providing them . It is the work of many others that has applied them successfully to fluids.
The flaws are there: the rigid motion similes, the misapprehension of the Twistorque and Twistorque vectors, the reliance on torque models; but as a functioning model it is very impressive.
The strain ellipsoid is a very useful tool, but when considering the propagation of strain , and of stress doing the straining, we clearly arrive at Maxwells elegant mosaic of vortices with slippery sphere se in between . Without the spheres we can imagine a much shorter and slower propagation of strain.
I may usefully employ the hyper or Hupostasis ytrochoids as images of strain propagation through such media.
Quantum mechanics of the Schroedinger wave equation is a study of the interpolations of the wave function. As Dirac varied the wave function through time, Dirac's study discusses how those interpolations change with variation. The classical mechanics provides the points from which interpolation starts. Extrapolation from the quantum wave equations to classical points of observation will fail if taken beyond those points.