Posted: Sat Oct 24, 2009 6:57 pm
Something of this kind should apply to the behaviour of the Polywell.DeltaV wrote:An example of a tensor that can be thought of as composed of dyadics is the inertia tensor of a 3D rigid body.
a discussion forum for Polywell fusion
https://talk-polywell.org/bb/
Something of this kind should apply to the behaviour of the Polywell.DeltaV wrote:An example of a tensor that can be thought of as composed of dyadics is the inertia tensor of a 3D rigid body.
The thing I pick up from reading stuff by quaternion boosters is that they tend to feel that (a) Heaviside & Gibbs dropped important stuff from quaternions in their work, stuff of real physical significance, and (b) that more modern work in GR and QM are based on Heaviside & Gibbs vectors, and are thus missing the dropped stuff from Hamilton.DeltaV wrote: If you can accept that Gibbs-Heaviside vectors (those commonly taught in high school and undergraduate college courses) are (more or less) derived from the more general quaternions (which are actually a pairing of a scalar with an "axial" or "pseudo" vector, pseudovectors being able to stand in for an ordinary or "polar" vector with no loss of information, but definitely not being the same thing), and ignore Gibbs' and Heaviside's denials that this borderline thievery actually took place, then I hope you can see that if a "flat space" restriction applied to quaternions it would also apply to ordinary vectors, hence to tensors. This does not seem to be the case, since tensors (presumedly based on dyadics) are often applied to curved spacetimes in GR.
It's really a confused mess that should have been hammered out over a century ago.
This is a straight contest between mathematical models. Some models are easy to test, some can only be tested during an unusual event like an eclipse, and some (like string theory) cannot yet be tested.blaisepascal wrote:When I hear people talking about how quaternions are so much better for EM, GR, and QM than Heaviside/Gibbs vectors, and how this should have been resolved a century ago, I feel like they are missing a century of mathematical physics.
Do you mean "real physical significance" as in traditional 3x3 rotation transformation matrices (the columns of which are the transformed G&H unit vectors) completely ignoring the physically demonstrable fact* that rotating by one full turn (or an odd number of full turns) about a fixed axis is topologically NOT equivalent to rotating by two full turns (or an even number of full turns)? Quaternions collapse to integer +1 for even numbers of full turns (strings not entangled) and to -1 for odd numbers of full turns (strings fully entangled), which corresponds to observable physical reality. G&H-based rotation matrices do not even address the intrinsic topology of rotations in 3D.blaisepascal wrote:The thing I pick up from reading stuff by quaternion boosters is that they tend to feel that (a) Heaviside & Gibbs dropped important stuff from quaternions in their work, stuff of real physical significance
See Sachs' books/papers, where he says quaternions are the building blocks ("irreducible representations") for the "most general" form of GR. He's not throwing out Einstein's work, but rather extending it to its full potential. Sachs' GR metric has extra, rotational aspects ("spin degrees of freedom"), and he asserts that QM falls out of this new formulation in the linear, low-energy limit.blaisepascal wrote:and (b) that more modern work in GR and QM are based on Heaviside & Gibbs vectors, and are thus missing the dropped stuff from Hamilton.
I agree completely.blaisepascal wrote:The flaw is that "vector", as a mathematical and physical concept, has been manipulated, generalized, and overused over the years to the point of confusion between the different meanings.
I think most of the trickiness about the "proper" quaternion results from the above-mentioned topological property (odd vs. even number of full turns) not being taught in high school, as it ought to be. Trying to express a rotation (or a rotation-dilation-entanglement - "Gravitation", Misner, Thorne and Wheeler, Ch. 41, I think) with a method that doesn't account for the topology means giving up some information, causing ambiguity. Taking an active rather than a passive view of rotation, when feasible, sometimes makes things easier too.blaisepascal wrote:Rotation is conceptually simple, but a tad tricky due to the issues of computing the proper quaternion for the desired rotation -- but combining rotations is easy, given the properties of a quaternion.
I believe that mathematics (at least the deep parts) is discovered by humans, not designed/invented, but if you want to think of it as the latter, go ahead. G&H vectors are intended for 3D. Agreed. I however disagree that quaternions are intended only for 3D. That was Hamilton's original intent, which failed, and he then realized he needed 4 dimensions (3 of one sort and 1 of another sort) for things to work out. On the family tree of mathematics, there are four allowed "division algebras": real, complex, quaternion, octonion (dimensions of 1, 2, 4, 8). The last two are non-commutative and the last one is also non-associative. It has been proven (1898) that there is no further progression of this sequence.blaisepascal wrote:Both Hamiltonian quaternions and Gibbs vectors are designed for and work with 3 dimensions, which was fine when the general belief was that physics was 3 dimensional (and Euclidean).
Agreed.blaisepascal wrote:SR and GR can't use Gibbs vectors because they don't work in (3,1)-space.
Wrong. Quaternions are very similar to spinors and Pauli spin matrices, but don't need any extra "i"s. Many researchers have used quaternions for spacetime since 1914. A few of the many more recent writings are M.J. Walker or J. Kronsbein (Am. J. Physics, 1950s-60s), Misner-Thorne-Wheeler (1973), S. DeLeo ( http://arxiv.org/PS_cache/hep-th/pdf/9508/9508011v1.pdf ). My own preference is for approaches that don't use complexified quaternions (aka biquaternions). Complexifying them makes them no longer a division algebra.blaisepascal wrote:Quaternions can't be extended to 4 dimensions either.
The goal should be to find the irreducible representations that give the most general expressions for spacetime physics, like Sachs does.blaisepascal wrote:The modern use of vectors is a generalization of the idea that a Gibbs vector can be represented as a triple of numbers. Instead of cross vectors, a rich panlopy of linear operators and tensors exist which accomplish the same goals, and more, and can be generalized to any number of dimensions.
Funny, that's how I feel when I hear them say the opposite.blaisepascal wrote:When I hear people talking about how quaternions are so much better for EM, GR, and QM than Heaviside/Gibbs vectors, and how this should have been resolved a century ago, I feel like they are missing a century of mathematical physics.
If quaternions can help with this, that would be a step forward.rnebel wrote:I also notice that a number of people are trying to make Polywell arguments using classical collision models. The dominant mechanisms for transferring energy between the ions and the electrons are collective mechanisms, not classical binary collisions. Our experience is that you have to do full-up kinetic simulations if you want to understand these mechanisms and their effects. We've been doing that for the past 1.5 years, and we plan to be doing a lot more simulations over the next 2 years.
A link to the Nebel quote:alexjrgreen wrote:If quaternions can help with this, that would be a step forward.rnebel wrote:I also notice that a number of people are trying to make Polywell arguments using classical collision models. The dominant mechanisms for transferring energy between the ions and the electrons are collective mechanisms, not classical binary collisions. Our experience is that you have to do full-up kinetic simulations if you want to understand these mechanisms and their effects. We've been doing that for the past 1.5 years, and we plan to be doing a lot more simulations over the next 2 years.
Something like this: Quaternion-based rigid body rotation integration algorithms for use in particle methodsMSimon wrote:A link to the Nebel quote:alexjrgreen wrote:If quaternions can help with this, that would be a step forward.rnebel wrote:I also notice that a number of people are trying to make Polywell arguments using classical collision models. The dominant mechanisms for transferring energy between the ions and the electrons are collective mechanisms, not classical binary collisions. Our experience is that you have to do full-up kinetic simulations if you want to understand these mechanisms and their effects. We've been doing that for the past 1.5 years, and we plan to be doing a lot more simulations over the next 2 years.
viewtopic.php?p=20032#20032
It's not just when they are energized. They have to see a changing acceleration, properly phased, while the energy is changing.UncleMatt wrote:I have yet to see proof that capacitor stacks actually gain mass when energized. I have been following this technology for many years, but can't seem to find clarity on that question. Please let me know if I have missed something, and if possible please point me to a duplicated research study where they found proof such a mass fluctuation is actually produced by capacitor stacks. That would increase my confidence in this concept actually going somewhere.
Molecular dynamics researchers also use quaternions quite often, and they've developed some interesting approaches to numerical integration for simulation.alexjrgreen wrote:Something like this: Quaternion-based rigid body rotation integration algorithms for use in particle methodsMSimon wrote:A link to the Nebel quote:alexjrgreen wrote: If quaternions can help with this, that would be a step forward.
viewtopic.php?p=20032#20032
Forgive my crude phraseology. My point remains, however, in that I have not seen enough proof in the form of duplicated experiments with direct confirmation that the transient mass fluctuations claimed actually exist that are the basis for this concept. I have read the papers about the experiments performed by March and Woodward, but still looking for experiments from their peers with supporting data and conclusions. I would love for it to be true, and make this concept practical, but still haven't seen enough evidence.DeltaV wrote:It's not just when they are energized. They have to see a changing acceleration, properly phased, while the energy is changing.UncleMatt wrote:I have yet to see proof that capacitor stacks actually gain mass when energized. I have been following this technology for many years, but can't seem to find clarity on that question. Please let me know if I have missed something, and if possible please point me to a duplicated research study where they found proof such a mass fluctuation is actually produced by capacitor stacks. That would increase my confidence in this concept actually going somewhere.
http://physics.fullerton.edu/Woodward.html
It's a direct consequence of e=mc^2 and Newton's second law...UncleMatt wrote:Forgive my crude phraseology. My point remains, however, in that I have not seen enough proof in the form of duplicated experiments with direct confirmation that the transient mass fluctuations claimed actually exist that are the basis for this concept. I have read the papers about the experiments performed by March and Woodward, but still looking for experiments from their peers with supporting data and conclusions. I would love for it to be true, and make this concept practical, but still haven't seen enough evidence.
I understand the theory behind it, I am waiting for experimental evidence to show that one can measure the mass variances with regard to this concept, and then in a practical way produce propulsion effects. I'm not against this reasearch at all, just looking for more verifiable evidence than is currently available, thats all.alexjrgreen wrote:It's a direct consequence of e=mc^2 and Newton's second law...UncleMatt wrote:Forgive my crude phraseology. My point remains, however, in that I have not seen enough proof in the form of duplicated experiments with direct confirmation that the transient mass fluctuations claimed actually exist that are the basis for this concept. I have read the papers about the experiments performed by March and Woodward, but still looking for experiments from their peers with supporting data and conclusions. I would love for it to be true, and make this concept practical, but still haven't seen enough evidence.
m= e/c^2
F = dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt)