Thursday, December 9, 2010

Unclamp it for Milgrom (Clamp it for Euler)

Cambride University's website has the following experiment: Gyroscope hanging over the top of a table.

Please note carefully that there are two versions of the same experiment on the page. The first experiment shows the unclamped gyro staying stable until the 12th line from the right. The third video with the clamped gyro shows the gyro system as being able to maintain balance only until the 9th line.

According to the good engineer's own description of the results "When the gyroscope is clamped so hanging out from the table the couple due to its weight causes the stand to topple....With the pivot point unclamped precession still occurs even when the stand is "upside down"

i.e.: An unclamped gyro behaves differently from a clamped gyro. Why? What's different about the clamped condition as opposed to the unclamped condition?

The extra advantage an unclamped gyro has over a clamped gyro is that it can sustain an infinitesimal net positive acceleration about an X-axis passing through the pivot point (with Y being the vertical axis and Z being the gyro's spin axis) - and its this situation that activates MOND - Modified Newtonian Dynamics. This is something which a clamped gyro is denied because the clamping causes the automatic transmission of even infinitesimal accelerations to the rest of the framework holding the gyro in an infinitesimal time.

Under such conditions, we must use the compound object concepts I have proposed to analyze the behavior of the unit. We can no longer consider the system under study as one object -rather we view it as two objects undergoing sequentially a collision and a decay process with a certain specific frequency we will call the Characteristic Frequency of the system.

Milgrom's boundary condition for MOND (Modified Newtonian Dynamics): Milgrom noted that Newton's law for gravitational force has been verified only where gravitational acceleration is large, and suggested that for extremely low accelerations the theory may not hold. MOND theory posits that acceleration is not linearly proportional to force at low values.

It maybe possible that MOND theory is applicable to infinitesimal systems with relatively large angular momenta suspended in an inductive state. (What is an inductively suspended angular momentum? That question is addressed in the rest of this Upaya- thought post and a direct definition of inductive suspension of spinning wheels is given in this link to my earlier Upaya from last year.)

The condition of an unclamped gyroscope in precession is analogous to Milgrom's boundary condition for MOND. The unclamped gyro in precession is able to possess an infinitesimal acceleration - something which a clamped gyro is denied. Under such conditions, we must use the compound object concepts I have proposed to analyze the behavior of the unit. This reveals that there is a way to resonate the energy transfer processes to navigate the gravitational field.

In order to properly analyze systems with angular momentum as the dominant player, we must first discover its suspension type to understand the behavior of the system. If the system has only capacitive suspension (the axis of the spinning object is secured firmly to an inertial frame containing the spinning object), then spin effects are minimal - if however the spinning object has some about of inductive suspension then the effects will be different and can be analyzed using the equivalent electrical loaded lines as an analogue.

For the rest of this Upaya-post and the next Upaya-post, I will focus on explaining how to construct an an equivalent electrical lines by first reviewing the development of lattice theory and analogies between electrical loaded lines and mechanical systems.

Background And Issues In Lattice Theory

Newton and Euler were among the original architects of a Lattice Theory of Space. In 1658 in the midst of a very unusually strong storm, Newton measured the velocity of the wind using his very own method involving performing long jumps with the wind on his back - comparing it to how much he could jump on his own power gave him the total force which when divided by the surface area of his body's cross section in the plane perpendicular to the wind velocity gives the pressure of the wind accurately.

Yet Newton also imagined that in the midst of all that turblulence, that air was in fact best modeled by imagining it as a continuous chains of point masses connected to their next nearest neighbors on either side (1-Dimensional model) by elastic springs. Newton adopted that model when he attempted to calculate the speed of sound. Newton assumed that sound was propagated in air in the same manner in which an elastic wave would be propagated along a lattice of point masses. He asssumed the simplest possible such lattice (as luck would have it, it was the only sort for which, he knew how to apply the newly invented calculus)- one where each point mass is connected to its neighbors on either side, along the line of propagation.

Taking the elastic force constanst to be e, the particles to be of mass m and the distance between the masses to be d, Newton calculated the

velocity of propagation, V to be d*sqrt(e/m) - sqrt(e*d/rho) where rho = density of air.

Newton then attempted to take the (e*d) term and substitute the isothermal bulk modulus of air in its place, possibly arguing that since the maximum displacement possible for the springs is d, the behavior of the arrangement resembles that of elastic material in Hookes Law of Elasticity (Hooke has just proposed his Law of Elasticity just 5 years earlier). History records that it was Laplace in 1822 who substituted the correct value i.e. the adiabatic elastic constant and first correctly computed the speed of sound.

Euler is credited with being one of the first to apply calculus to physics. He carried Newton's work forward with his theories. He constructed his own theory of light and its propagation by analogy with sound. Now, among several theoretical leaps to occur as a result of such an analogy, there was also an issue: The theoretical models implied longitudinal vibrations - obviously contrary to later findings that light and all Electro-Magnetic waves are transverse in nature, but it is nonetheless informative. This important problem tells us that the scientists who originally grappled with the problem of energy propagation intuited energy as transmitted through the perturbation of the lattice's elastic space. The matter points might be viewed as the nodes of a string instrument and the elastically bound fields then vibrate back and forth and the stretched string sets off waves along a line of points of matter connected to their two nearest neighbors by elastic springs.

Might I also remind the science historians, Eulerian theory of colour accounted for clolours by means of a specific resonance excitation;

Euler explains colours in the following way: the ray incident upon a surface or the matter of light hitting the surface, puts the smallest particles of this surface into vibration. The more elastic the particles, the quicker the vibrations proceed are so that the same ray produces a different number of vibrations per second. The number of vibrations per second determines the colour like that of a string determines the tone."

Euler is invoking here, the second law of vibrating strings which states that

Second Law :
When the length ( l ) and the linear density ( m ) are constant , the frequency of vibration ( n ) of a stretched string vibrating in one segment is proportional to the square root of the tension ( T ) in the wire .
i.e n ∝√(T)
or n/√(T) = constant

Euler was thus viewing the distance between each successive particle within the matter of the surface reflecting the light as having the same role as the length of a stretched string on a musical instrument and each set of two nearest neighbors have the same role as the two nodes of the string producing one single wave between them. In this fashion, Euler subscribed implicitly to a lattice theory for all energy generally.

Baden-Powell's treatment of Newton's model and its application to a cubic lattice represents the first successful mathematical analysis of a 1-D elastic wave. He computed the velocity of a wave propagating along one axis of the cubic lattice structure as a function of a which is defined as (1/applied wave length).

The figure below shows the distribution of the wave velocity against a, the inverse of the wavelength of elastic wave. As the figure shows one major factor is d, the distance of the point masses from one another in the lattice. Baden-Powell's equation for the propagation velocity V of the elastic wave is V = Vinfiniti * |sin pi()*d/lambda|/(pi()*d/lambda).

Take Baden-Powell's velocity formula above. Substituting v infiniti/nu = lambda, we get a constant velocity wave i.e. a wave that travels at a constant speed for example a light wave or a sound wave with a medium. All the other waves have a velocity dependent upon the wavelength implying that except for a completely monochromatic source, the wave would become diluted in space quickly due to the different distances covered by non-monochromatic waves, where as a completely monochromatic source is able to proceed as a soliton. It was Kelvin who gave a complete treatment of 1-D elastic waves. Kelvin assumed the same lattice as Newton and Baden-Powell and numbered it as shown in the following diagram.

By the time we go from Baden-Powell in 1841 to Kelvin in 1881, the lattice model has transformed from suffering longitudinal movements to transverse movement. This change was necessitated by experimental evidence of the transverse nature of most waves and which made it possible to include frequency but also in my opinion, brings about the Size Problem. The particles' transverse movement made it possible to sustain an analysis that held the lattice spatially stationary and of constant volume and shape during the wave propagation - a condition which fit the vast majority of wave interactions - while the longitudinal models fit only piezoelectric crystals.

The Size Problem: In Kelvin's models, significant numbers of point masses can rise upward and fall downward about their mean position. The question is how can the entire chain of masses be really continue to stay within their allocated maximum travel distance, d if you assume that a transverse movement takes away longitudinal length. Since each point mass's loss of longitudinal length will add to the rest, the entire chain must exhibit significant, measurable size changes between its vibrating and non-vibrating states. By the laws of probability, in a vibrating crystal or material of any kind with a large number of point masses, far more point masses are in non-equilibrium positions, as opposed to the equilibrium position of which there is only one, If a vast number of pointmasses are in nonequilibrium positions this implies that the crystal should visibly change size (the large the crystal, the greater the net change in the size of the crystal). Piezoelectric crystals for instance show measurable dimensional change. However a vast majority of interaction do not show such dimensional change.

The way we currently view Kelvin's lattice model (and all other newer models in modern lattice theory) is that we assume that the transverse movement of the point masses happens a seperate dimension whose gain will not be the longitudinal dimension's loss, and yet is somehow determined by d, the longitudinal wavelength.

Take this specific problem of how the model used by Kelvin so successfully later on to model all manner of waves, nonetheless throws up this issue: The original longitudinal model has been replaced by a transverse model because it works. However, there doesn't seem to be a reconciliation done of how it is possible that there is no size change in such a model along the longitudinal direction? What happened to it? Its not there, so its ignored. Its the dog that didn't bark.

One can reconcile the Size Problem of the transverse wave model by accepting that a large amount of inductively suspended spin angular momentum would serve as the fundamental unit of a SpaceTime lattice. Like a gyroscopic problem, the mechanism inside will then seem to have somehow orthogonalized the energy input and output channels, thereby allowing for transverse waves. The lattice model survives, but we no longer see the point masses as moving logitudinally because the gyroscopic effect of the Spin units causes the input and output channels to be orthogonal and therefore the absence of longitudinal reaction to the advance of a longitudinal wave is not a show stopper. Conventional physics doesn't really address these issues currently. It may be that the all interactions involving transverse waves are ultimately all mediated through gyroscopic exchanges of energy between nearest neighbhors possessing large quantities of inductively suspended spin angular momentum.

Much of what is in this post is derived from the work of Leon Brillouin's Wave Propagation in Periodic Structures. I am indebted to him for fantastically original works that make the inner workings of nature crystal clear to the student. Another minor source is the book "Discoverer's of Space" by Erich Lessing. *Also cited is a statement of the first law of vibrating strings from and Euler's opinions from Leonhard Euler: Beitraege zu Leben und Werk by Johann Jakob Burkhardt. Also cited is the wikipedia entry on Modified_Newtonian_dynamics .

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