Classical mechanics is an impossible trick. The resolution of Zeno's paradox assumes Lorentzian invariance and the relativity of space and time, first, and that you will believe the fanciful stories physicists cook up, second. I do not, which is why I dare not move.

Faster,
liberalists,

calculus or some such thing solve Zeno?
I thought that you just had to add up the halves e.g. what does this sequence sum to:
1/2 + 1/4 + 1/8 + 1/16 + 1/32
[ Parent ]



Zeno's four arguments against the assumed existence of continuous, infinitely divisible space and time depend on the convergence of geometric series.

Faster,
liberalists,
Parent ]



...this raises a very intruiging question.
Do you have any knowledge pertaining to matters outside the field of Geometric Algebra?
Yicky Yacky
***********
"You f*cking newbie. Shut up and sit in the corner!"  JCB



but I can only write just so fast. 8)
The people nearest to me who come in range of my speech know I'm full of something (else).

Ask me about space and why we must treat it as the next frontier to be opened and I'll get downright evangelical.
Dream Big.
Grow Up. [ Parent ]



Ask me about space and why we must treat it as the next frontier to be opened and I'll get downright evangelical.
You'd be preaching to the perv converted...
Yicky Yacky
***********
"You f*cking newbie. Shut up and sit in the corner!"  JCB [ Parent ]



Fresh crop of... OpEds (none / 0) (#27)
by adiffer on Sat Jan 25, 2003 at 12:10:58 AM EST


We should start a new series then. It should relieve the Iraq pressure people are feeling right now to look a little further into the future than 'Who's next?'
Dream Big.
Grow Up. [ Parent ]



It would be a refreshing change from drivel.
You can show a rock how to jump only so many times before you give up believing that rocks can jump.  K. B. Salazar [ Parent ]



is not helping. I was trying to type:
It would be a refreshing change from the drivel that sometimes passes for articles. This series on Geometric Algebra is superb. A few pieces on Space would be almost like dessert. Brain candy? ;)
You can show a rock how to jump only so many times before you give up believing that rocks can jump.  K. B. Salazar [ Parent ]



A very good idea.
I'm not sure where a good place to start would be though. The topic is so vast.
One the facets that always fascinated me was the episode regarding Gary Flandro (and earlier work by Michael Minovitch).
It becomes slightly teleological, given that there were five or so years between the 'discovery' and the commencement of serious work on the project (in which time somebody else may have noticed), but it always tantalized me that, but for one parttime student who'd been essentially told to stop pestering those on the lunar mission, and to 'bugger off and do some maths somewhere while we concentrate on important stuff', we would not have half the data (visual and otherwise) we have now.
Raw visual imagery became so important for communicating the missions' significance to the outside world and the human race (several key scientific discoveries notwithstanding), that I can't help wonder whether the lack of engagement by government(s) and the general public is due, in part, to the lack of 'direct' experience currently being provided / provoked. Hubble has been hugely beneficial, but perhaps has been partly responsible for a lack of enthusiasm for direct engagement. 'Why bother when a semistationary eyeinthesky can seemingly provide so much?' etc.
I can't help feel that that there is a recursive feedback mechanism at work with regard to the opinions / enthusiasm of the general public and government(s): Whilst engagement and exposition brings greater enthusiam, which brings further engagement and exposition etc., the converse is also true, and appears to be what might be happening currently.
The lack of an 'adversary' also seem to be a problem. Perhaps with China and Europe beginning to make more and more adventurous forays, this will reverse the status quo and accelerate the feedback loop in the opposite direction.
PS: You seem to be working in the right place ;D... Perhaps you are in a better position to provide the more scientific viewpoint than I am. I will have to think on it. I am more 'fascinated layman' than seasoned pro', and suspect any contribution I might make would be along the lines of a more philosophical 'for the good of the species' argument than any other.
Yicky Yacky
***********
"You f*cking newbie. Shut up and sit in the corner!"  JCB [ Parent ]


One should not vote +1 on articles one does not understand because they are hard science.
em



I get the feeling some might be doing that, but quite a few are not.
I believe my target audience is composed of many people who want to learn something new. Whether it is hard science or soft whatever, they like feeling that they are becoming more knowledgeable every day.
If I'm right, they are not voting on the factual content of my work. Instead, they are voting on a style and a belief that they will benefit later once they have had a chance to absorb the content. If I'm right, discussion will be minimal and delayed.
Dream Big.
Grow Up. [ Parent ]



Two questions.
 If I wanted to learn any of this stuff, why should I not instead of reading your article, go pick up a book among the countless that have been written on it, in countless styles and levels of detail and/or assumed background, best matched to my interests and needs?
 In light of this, what does your article have to offer to the readers of a discussion site, other than excusing the intellectual laziness that prevents them from going to a library and picking up a few books? (Oh, and nobody answer "not everybody has time to read this stuff"; because the answer to that is "stop wasting your time on k5, then".)
em [ Parent ]



1. By all means go right ahead. I have been doing just that, but if it weren't for ADiffer's articles, I would have never heard of geometric algebra in the first place. (Thanks, ADiffer)
2. This site is not only for discussion of people's opinions, but also for introducing topics which at least one person would like to see a lively, and hopefully intelligent discussion. ADiffer is posting this because he believes that it may be of interest to some segment of the k5 community (which it is).
In light of all this, what does your post contribute to what should be a discussion of Geometric Algebra as applied to Classical Mechanics?
Ad Astra Per Aspera [ Parent ]



If you can find a book anywhere that lays out this subject the way I do, I would be very surprised. I've looked far and wide and not found a single one.
What you are seeing here is original work. You (or others) are benefiting from seeing a theoretical physicist do what we do when we our work. I am creating for you all.
You have a point about this being a discussion site, though. I wasn't going to post any of this work until I saw the particle physics review series and the response it received from K5 readers. Apparently, K5 is more than a discussion site.
Dream Big.
Grow Up. [ Parent ]



I agree with ADiffer that what we are reading is an original work on his part. He is attempting to manage the difficult feat of writing for the layperson. I am fortunate to be a little more technically minded and constantly curious, so I've sought out many other references on the subject. I would recommend either of the following references as starting points to anybody interested enough by ADiffer's articles to want to known more details.
David Hestenes' book, New Foundations for Classical Mechanics, does a very good job of both introducing Geometric Algebra and applying it to both Classical and Relativistic Mechanics. His writing style is very readable, but gets bogged down in the math every once in a while (but hey, that's the nature of the material).
I've also found that the lecture notes at the Cavendish website are also a good place to get started. The notes start out easy to read, but the pace picks up pretty rapidly after the first couple of chapters. The main feature of these notes is that they take you all the way through classical and relativistic mechanics, as well as quantum mechanics and a couple of other advanced physical theories. The notes are rather brief though, so don't expect to come away with a perfect understanding of everything they present.

AD, it looks like we're keeping the same hours. :)
Ad Astra Per Aspera [ Parent ]


These days all physicans believe that classic mechanics is fundamentally flawed to the "residual gap".
It's well known that
rot F * A = integral div V cross PL d mu + nabla phi.
This means that nabla phi is a residual operator. But classical mechanics say that is constant zero which is obviously wrong.
You too can become a Mensa member !



And you have forgotten at least one thing too ... (5.00 / 1) (#37)
by dougmc on Sat Jan 25, 2003 at 02:37:52 PM EST


These days all physicans believe that classic mechanics is fundamentally flawed to the "residual gap".
`all' is an unlikely number. `Most' would be much more likely. More on this later.
But what I really wanted to point out is that this is `classical mechanics'. It's believed to be mostly correct when dealing with macroscopic entities and speeds much slower than the speed of light. When you start dealing with the microscopic, quantum mechanics becomes important, and when you start approaching the speed of light relativity becomes important.
Modern science `knows' (i.e. most people versed in modern science know/believe) that classical mechanics do not completely describe the universe. However, it has shown itself to be incredibly useful, over and over and over, for predicting how things happen without too much work. As long as you restrict yourself to items that are much larger than individual atoms, and much slower than the speed of light, it's an almost perfect approximation, and this is not going to change, no matter how much science progresses.
I have a B.S. degree in Physics, and I still found the article interesting.
Does the B.S. degree qualify me as a `physican' ? If so, I do not find classical mechanics to be `fundamentally flawed'. It has it's limits, but within it's realm, *it is how* we calculate how the world around us works. Sure, we *could* use quantum mechanics and relativity to calculate the trajectory of a baseball, but why? It would give the same answer, and only require a million times more work.
[ Parent ]



Classical Mechanics does work quite well as you pointed out for problems it was designed to manage. Small, fast, and other wierd situations show its flaws.
Beyond all that, though, there is also its value as a theory about reality. Anyone learning to be a theorist must learn how the old ones are constructed, where they break, what the underlying written and unwritten assumptions were, and how to roll their own. Classical Mechanics is a well tested tool that has seen translations from one math language to another at least twice so far. It has also been abstracted to such a level (Action Principle) that modern users have a difficult time recognizing early versions and terminology.
To a theorist, Classical Mechanics is a gem whether it works well in any one situation or not. The metaknowledge for this branch of Physics is deep and rich. Anyone wanting to improve their skills as a theory maker must get this kind of metaknowledge wherever they can and Classical Mechanics is the oldest source.
Dream Big.
Grow Up. [ Parent ]



Excellent article on Newtonian Modeling (none / 0) (#46)
by PhysBrain on Sun Jan 26, 2003 at 04:27:47 AM EST


Speaking of metaknowledge, I recently finished David Hestenes' article entitiled Modeling Games in the Newtonian World. It is a very interesting look at the underlying assumptions and goals of physicists and scientists in general. His premise is that "in science, modeling is the name of the game", and "The object of the game is to construct valid models of real objects and processes."
The purpose of the article is to serve as motivation for changing the way physics is taught, but there is alot of other really good stuff about the history of Newtonian mechanics and its evolution to its current form. The seven assumptions, the Zeroth through Sixth Laws, stated by ADiffer in the beginning of his post are first (to my knowledge) introduced here.
Ad Astra Per Aspera [ Parent ]



I encountered various forms of the other laws besides Newton's three in other places, but Hestenes' version was the first one I read that brought them together in a coherent way. I didn't find them in the paper you listed, though, so I'll have to give that a read.
In the first edition of his book 'New Foundations for Classical Mechanics', he spells out the purpose of modelling and those laws/assumptions. I've made a few variations to account for the fact that I expect all my physical objects to be multiranked, but that doesn't really change the metarules for modelling. When I read chapter 9 of that book, I wondered why he had not taken advantage of the full power he was discussing in chapter 9 when he wrote his earlier chapters. Had he done so, he really would have created a 'New Foundation'.
My work tends to use the modelling rules set out by Hestenes and the object rules set out by my graduate advisor Ken Greider. You won't see much of Ken's stuff in print, though. I have it because I inherited the file cabinet with his notes when he passed on.
Dream Big.
Grow Up. [ Parent ]


Remember, decimal ruins everything. (1.12 / 8) (#40)
by Fen on Sat Jan 25, 2003 at 05:57:56 PM EST


At some point you need to put numbers down to measure. Use hexadecimal then, never decimal. Decimal represents ugliness and cancer. Hexadecimal is elegant and beautiful. Metric works with any radix, so use hexadecimal. Self.


I think classic newtonian physics is good stuff. But I just had this thought. A basic newtonian rule states that a body in motion stays in motion unless acted upon by another force. So in theory a baseball thrown in space would keep going forever as long as it didn't hit some dust or wind. We will forget the fact that no matter how far away another object is in space, it will still have a gravitational pull on said baseball, even if it is so small it isn't measurable. Now think about this, a simple rule of quantum mechanics says a system cannot be observed without altering its conditions. I think I pulled that out of the Schrodingers Cat example.(I was about to say nothing can be observed without altering its conditions. All the progammers woulda gone, "yup.") So even though newtons "bodies in motion and bodies in rest" rules make perfect sense, we can never test the theory so it cannot be proven.
I just thought I'd share that crap with you.



The effect is studied and is known as the Quantum Zeno Effect. It's basically a reassertion of the AWatchedPotNeverBoils truism. To wit:
1) Systems, once observed, fall into an eigenstate of the Hamiltonian.
2) Once in a true eigenstate, the state evolves in phase but not into any other eigenstate (the working definiton of an eigenstate). In the presence of a *weak* perturbing Hamiltonian that might mix states, the flow of probability into other states is linear in time to first order (e^(iHt} = 1 + iHt + ...) , so...
By continuously observing (or by approximating constant observation), one should be able to force a system to stay in one eigenstate forever. And it's been done experimentally in atomic systems; do a quick paper search. There is, in some systems, a corresponding AntiZeno Effect. Nothing wacky going on, just the statistical factor of Fermi's Golden rule coming into play.
[ Parent ]



You could read it differently, I guess (5.00 / 1) (#55)
by inerte on Sun Jan 26, 2003 at 09:36:27 PM EST


You said: a body in motion stays in motion unless acted upon by another force.
My guess is that it isn't wrong to say: a force can modify motion. By stating that a system cannot be observed without altering its conditions we are implying that if we are not observing it, it isn't changing.
Perhaps the abscense of proof is what makes it true. Can such thing exists? ;)
Ps: Classical example: If a tree falls in a forest and there's no one there to hear it, does it make a sound?

CID 4596201: Of course power users can always use another distro, or just [ Parent ]



we are implying that if we are not observing it, it isn't changing.
Thus it is in an unknown state. And that state will not be met until we observe the sytem. So what's in there while we are not looking?
[ Parent ]



I had a nice long reply for you the other night and then it got eaten by the system or an upgrade or something.
You are on an important point. An unspoken assumption in the Newtonian model that probably should have been included in the list of laws involves absolute precision and a form of omniscience. Experiment has shown both parts to be false and we were forced down the quantum/probabilistic path.
Anytime someone can say 'But you can't test that part' we should all take a step back from our model no matter how much we love it and fix it. An untestable part in a model is usually a sign of a major flaw associated with an unspoken assumption.
Dream Big.
Grow Up. [ Parent ]


Zero Law: Objects have a continuous history in space and time
Can you give me an example of such an object, that has a continuous history in space and time?



The zero law is one we require for our kinematics variables to make much sense. A continuous history is assumed to ensure the velocity and acceleration definitions work in the instantaneous sense. This law basically translates to mean that objects can't magically disappear from one place and appear in another.
I mean, objects always magically disappear from one position and appear to another, while they are running. Well, most of they times they appear to the closer position, but this does not mean that the do not magically disappear from the previous place.
[ Parent ]



The zero law basically demands smoothness for the variables that describe motion. The easiest way to understand that without calculus is to see when the assumption is broken.
Imagine you were some comic book hero that could teleport from one place to another. When you did, you would be jumping from one location to another without traversing the region in between. This would be a break in smoothness and would break law zero. The discontinuity in this case is with L and it leads to our inability to define your V for that jump.
Imagine you were some different comic book hero and could make objects move with your mind. If you could change an object's speed with no time elapsed, V would be discontinuous and A would be undefined. This too would break law zero. If your act of telekinesis actually required a bit of time and the object had to pass through other speeds to get to the one you intended, law zero would remain intact.
Most people assume law zero is a good one. It gets tricky to maintain it, though, in a quantum theory. It can be done, but you have to change how we represent those objects by quite a bit.
Dream Big.
Grow Up. [ Parent ]



If you've had a course in calculus, think of the Fundamental Theorem of Calculus as your guide to continuity.
If you don't know what this means then think of an object's path through space being represented by a string. As time passes, the object moves along the string. A continuous history means that the string cannot be cut, and therefore the path that the object must take is not allowed to "jump" from one location to another.
This probably would make more sense from a functional perspective if ADiffer had choosen to add a time axis to his kinematic model rather than relying on the scalar term of L to "label" the time parameter. However, the introduction of four dimensional spacetime is not very intuitive to most people, so I can understand why he chose to represent time the way he did.
Ad Astra Per Aspera [ Parent ]



Care to guess what the distinction between time and proper time is going to be in location space when we shift up to R(3,1)? 8)
My current inclination is to make a break and use different ranks for each of them. I can't think of a good reason yet for why proper time has to be more than a label.
Dream Big.
Grow Up. [ Parent ]



For the rest frame (ie. location space), time would be the fourth axis of your system with the negative contraction rule. Proper time in the fly's reference frame will depend upon the history of his motion. However, it should be possible to make a transformation from the rest frame to the fly's frame on an instant by instant basis. That is to say that it should be possible to find the instantaneous time dialation in the fly's reference frame and length contractions in the rest frame based on the fly's instantaneous velocity.
Therefore, if you really want to label the fly's time coordinate with the scalar component of his position, be my guest. But it was my interpretation that the main objective of using multiranked vectors (ie. multivectors) was so that physically meaningful relationships could be obtained when multiplying multivectors using the geometric product. By careful selection of how you represent different parameters in the system you could come up with a completely consistent and very compact formulation for the model of a given physical phenomenon.
This is primarially a philosophical point, but if you have a good idea about how your representation is going to come together in the end, or if you plan on modifying your model later to demonstrate how to build a selfconsistent model, then I will await your further posts with much interest.
Ad Astra Per Aspera [ Parent ]



I'm currently thinking I might split the role of label for motion from the time direction and keep both around. I may change my mind later if I find the idea is a bunch of **, but right now I think there is something important behind the concept of proper time that isn't getting the attention it deserves.
(Usually that is a sign of my partial understanding of something a professor worked hard to drill into my brain and didn't quite get it there.) 8)
Dream Big.
Grow Up. [ Parent ]


I have decided that 'rank', in the context, means a cartesian coordinate, like X, Y, or Z. I could be wrong, but in the article, he never defines the term. I believe these articles are meant to stand on their own, without having to read his other articles first... is this true, or false?
It would be nice if he explained what the prerequisites are for reading his tutorials if they are not meant to stand on their own. And if they are, they reader would be better served with less undefined jargon.



Rank and grade are used synonimously most of the time. If you've read any of ADiffer's previous posts, then you would know that rank refers to the dimensionality of a particular directed quantity. Scalars have a rank of zero, vectors (ie. directed line segments) have a rank of one, bivectors (ie. directed plane segments) have a rank of two, and trivectors (ie. directed volume segments) have a rank of three.
Higher ranked objects are possible up to and including the dimension of the space you are trying to model. In the context of this article, only three dimensions are being modeled, so the trivector is the highest ranked object possible.
Ad Astra Per Aspera [ Parent ]



The author is essentially recasting physics, model by model, in terms of Geometric Algebra. Since this is a new mathematical tool for most of us, his first articles in this series introduce Geometric Algebra. So they are necessary for the understanding of what is going on.
In this and the previous article (on kinesthetics using Geometric Algebra), he is choosing to use an algebra known as R(3,0), which means it's an algebra with three "spacelike" coordinates and no "timelike" coordinates.
R(3,0) is defined in relation to three different orthonormal basis vectors (called e_{1}, e_{2}, and e_{3} for convenience), and a realvalued field (with an identidy element called e, in this formulation of GA). All the normal rules of vector spaces apply, with the addition of geometric multiplication.
Geometric multiplication is defined so that the product of two parallel vectors is a scalar value (i.e., no vector component) and is commutative. The product of two perpendicular vectors is a "bivector", or directed area, and is anticommutative. The product of three mutually perpendiclar vectors is a "trivector", or a directed volume. Products are associative. The general product of two vectors is a mixture of a scalar and a bivector, and a general value in R(3,0) is a mixed sum of scalar, vector, bivector, and trivector parts.
Given vectors as a linear combination of the three basis vectors, one can multiply them as if they were polynomials, keeping in mind that vector multiplication is, in general, noncommutative, so the order of multiplications needs to be correct, and the following simple simplification rules: e_{a}e_{b} can be writen as e_{ab}, e_{aa} = e, and e_{ab} = e_{ba}, for all a, b not equal to each other.
Bivectors in R(3,0) form a vector space with three basis vectors. It should also be noted that, given nonequal a, b, e_{ab}e_{ab} = e_{abab} = e_{abba} = e_{aa} = e, so every unit bivector is a square of e  i.e., bivectors act like imaginary numbers. And, like imaginary numbers, they are geometrically associated with rotation. Multiplying a bivector and a vector that lies in the plane of the bivector has the effect of rotating and scaling the vector in the plane of the bivector. As such, physical quantites dealing with rotation (like angular position, angular velocity, moments of inertia, torques, etc) get associated with bivectors. The product of two perpendicular bivectors is another bivector, in a manner analogous to the three imaginaries i, j, and k in quaternion mathematics.
Trivectors in R(3,0) form a vector space with one base vector (e_{123}). Squared, this base vector yields e, so it is not "imaginary" in the same sense as the bivectors. Trivectors, like scalars, commute with vectors and bivectors. When multiplied with a vector, it yields a bivector perpendicular to the vector, and viceversa, in a onetoone relationship. The product of a vector or a bivector and the unit trivector is called the "dual" of the original vector or trivector.
A typical value in R(3,0) is a linear combination of a scalar, a vector, a bivector, and a trivector, and thus has four "ranks", 0 through 3, depending on how many vectors are participating in the product. One can associate different physical quantities to each of the four ranks, if you so choose.
In the treatment of kinesmatics, the author chose to represent position L as a geometric object with time encoded as the scalar component, linear position encoded as the vector component, orientation encoded as the bivector component, and the volume as the trivector component. The velocity V is dL/dt, so the scalar component is 1, the vector component is the linear velocity, the bivector compont is the angular velocity, and the trivector is the "divergence" of the object being measured. A third quantity A is defined as dV/dt, and you can figure out what the various components are.
I think that should bring you up to date, but you really should read the rest of his series.
[ Parent ]



trivector is imaginary in R(3,0) (none / 0) (#67)
by adiffer on Tue Jan 28, 2003 at 03:15:35 AM EST


e_{123} e_{123} = e_{123123}
=e_{112323}
=e_{2323}
=e
Just a detail, though. Good summary otherwise.
[I had a mistake sitting in the multiplication table for R(3,0) for some time until a reader pointed it out. It's so easy to miss a minus sign.]
Dream Big.
Grow Up. [ Parent ]


1: How would the fifth law fall apart if one had to rely upon light traveling at a finite speed in order to know when two events are simultaneous?
Observers would agree that two events occurred simultaneously only when the light from each event arrives at their position at exactly the same moment. Imagine throwing two rocks into a lake. The expanding wavefront represents the propagation of "notification" of the event of each rock entering the water. When both rocks hit the water at the same time (with respect to the lake's reference frame), then the only observers on the surface of the lake who would "notice" both events at the same time would be those who are equidistant from both locations where the rocks entered the water. In general, the locations for observers who notice the events as simultaneous can be mapped out by plotting where the concentric circles of each wavefront intersect.
In the Newtonian world where time and length are both fixed for all observers, the Fifth Law would completely fail even for inertial observers in the same reference frame (at rest with respect to one another) except in the case mentioned above. For inertial observers in relative motion with respect to one another, there can be no agreement on simultaneity.
The special theory of relativity restores the fifth law by allowing the passage of time, and the distance between locations to change such that simultaneity is restored for all inertial observers. This site
has excelent visualizations of this principle.
Ad Astra Per Aspera



Our choice in that situation is to sacrifice law five or the algebra upon which we built all this. Like most good physicists, we sacrifice the solution space before we give up the structure that defines our modelling process.
The curious side effect of this is that we are more willing to change our definition of the universe than we are of altering the infrastructure we build for the model building process. 8) [...and it seems to work!]
Dream Big.
Grow Up. [ Parent ]


Looking forward to the next article....
The standard formulation of classical mechanics defines work or energy in terms of force and distance, or torque and angular displacement: W = integral{F dot ds} or W = integral{T dot dtheta}.
Since in your GA formulation of classical mechanics, forces and torques are combined into one object F, and distances and orientations are combined into one object L, I am assuming that work is going to be defined as a scalar function on F and L.
The obvious choice of a function would be integral{ F dot dL}, using the geometric product, but I'm uncertain how good of a choice that is. Does that product end up scalar? I'm not 100% certain that it does. It seems to me that you would end up taking the inner product between (say) the linear force and the angular displacement, which would give a vector result of some sort.
If you only took the dot product of the corresponding terms of equal rank, I can see getting a proper energy out of it. I'm not certain of the significance of the third rank, but it seems likely there is an energy component in it. The zeroth rank falls out because the zeroth rank of F is 0.
Am I on the right track for defining energy?



You are on the right track.
The traditional work done along a short path would be dW = SP( F · dL ) where SP() is the operation that leaves only the scalar part of a multivector.
However, whenever you see SP() it is time to be suspicious that we are missing something potentially useful in the other ranks. Accepting this definition would leave us violating our assumption regarding the higher form of covariance from the last chapter too.
I have to address energy in the next section, so you are heading in the right direction in another sense too. 8)
Dream Big.
Grow Up. [ Parent ]


Think about balloons. When you blow one up or let air out, they experience a volume changing force. The units would have to look something like kg meter^{3}/second^{2}.
If you take one of these forces and apply it across an area (divide by m^{2}) you should get the linear acceleration of that surface, right?
Dream Big.
Grow Up.



I had been thinking along those lines as well. The force per unit area would be pressure then.
I was also looking for ways to interpret the rank three component of M as some kind of density, and of P as a timedependent variation of density. Applying the notion of relating kinematic variables to dynamical variables through a proportionality constant that is the inertial mass one would hope to come up with a meaningful interpretation for the m*V and m*dV component. Unfortunately, it does not have the same kind momentum interpretation as the other two ranks. Unless, you consider a fixed volume element and allow the mass in the volume to change. Then the third term would be some form of mass flux or change in density over time.
I'm still trying to work on this idea. My thoughts are still up in the air. Where I work, we do alot of work with CFD simulations using solutions to NavierStokes equations on curvilinear and unstructured grids. I'd really like to see if there is an alternative GA formulation for the fluid flow models. If GA is really as coordinate frame agnostic as the literature suggests, this could be a significant contribution.
Ad Astra Per Aspera [ Parent ]



The rank three component in an R(3,0) algebra can't be reoriented in any way except through scaling. Your mass flux is probably close for P.
I'm inclined to think that a good GA library could do a lot of good for everyone with calculation intensive applications. That's why I wanted to write one for my own use in predicting solar sail orbits experiencing small perturbations from other sources besides photons. I'm pretty sure my library isn't good enough yet since my attempt to handle curvature is a bit weak and my reversion() method is buggy.
If various objects are recognized as parts of the same thing like I tried to do with L, I do believe the compactification of information and a good GA library will produce multiplications that are faster to run than the older matrix approach. I may be wrong, but I'm betting otherwise. I know I need a good computation person to recast my library, though.
Dream Big.
Grow Up. [ Parent ]



I've been experimenting with developing a GA library for a couple of months now, but I haven't gotten very far. The hardest part is trying to keep the coordinate frame independence of the blades. The only solution I've come up with so far is to just bite the bullet and use a Cartesian basis set for the original vector space V^{n}. Then I can define the blades for the multivector space as vector monomials. That means that I can still define an arbitrary basis sets for the multivector space, but the blades have to fall back on the Cartesian coordinate frame to get their relative orientations when interacting with other blades. This is not exactly a satisfactory solution, but it works for the time being.
I'd appreciate the opportunity to discuss library design with you some time.
By the way, in case you haven't been back to PlanetMath recently, I've been trying to come up with some GA entries for the encyclopedia. I've only got one up so far, but I'm working on a few more.
Ad Astra Per Aspera [ Parent ]



It has been a couple of months since I was there. I got a little busy with the rockets and stuff. 8)
One of the things I am supposed to do is convert the first three articles that focus solely on the geometric algebra to latex and put them up over there. Since I'm a newbie when it comes to latex, I face a learning curve.
I see your mail over there now. I'll get to the task of answering it.
Dream Big.
Grow Up. [ Parent ]


2: Suppose the fourth law wasn't quite true and an object experiencing two forces didn't act as if one force acted that was a sum of the other two. What impact would this have on the third law? Would the second law still be expected to hold?
I racked my brain on this for quite awhile. At first I was trying to conceive of a universe where the superposition principle wasn't true, but try as I may, I couldn't imagine a situation where two forces did not produce an observed motion in accordance with this principle. I guess this shows how biased my background has made me, but I always considered the superposition principle to be less of an axiom, and more of a theorem. That is to say that the superposition principle could be derived from the other laws.
I think I remember reading somewhere that the Third Law is actually known to fail when the Second Law is phrased as F = ma. I'm pretty sure it had something to do with moving charges and forces induced by their magnetic fields. This conflict is resolved if the stronger form of the Second Law is used, namely F = m(dP/dt).
Then I began thinking about the third law. There is an implicit assumption somewhere in there that the sum of all forces in a closed system should sum up to zero. Therefore, if the superposition principle could not be relied upon to be true, then the sum of all forces in the system might sum up to something other than zero, and hence the Third Law could not be expected to hold either.
Finally, this lead me to a case where the Fourth Law would not hold: noninertial reference frames. As I mentioned above, an implicit assumption in the Third Law is that all forces in a closed system sum up to zero. If there were an external force present, or we were in an accellerating reference frame, then the motion of the objects in the system could not be accounted for by the forces in the system alone. This would not necessarially break the Second or Third Laws but the external force would need to be accounted for in the end.
Ad Astra Per Aspera


3: Suppose you observed two large balls rolling toward each other across a frictionless floor. After the collision you note that one of them did not accelerate at all and the other one did. This observation would break law three. Would it break any other laws we described earlier? Explain.
If the Third Law is violated, then no known forces in the system can account for the observed accelleration. This would also imply a violation of the First Law which states that objects in uniform motion will remain in uniform motion unless acted upon by a force. Of course, the Second Law would no longer apply either since it explicitly links acceleration to applyed forces.
It is possible, however, to explain the observation without violating any of the laws. For instance, the difference in masses of the two objects could be so great that the path of the more massive object is not noticably altered by the collision with the much less massive object. Or, another force could have acted on only one of the objects, such as a very fast moving object which struck one of the objects and left the area before being observed. It's even possible that a complex internal mechanism, such as a gyroscope, could have been used to alter the path of one object before it struck the other. Only one of these explanations actually involve a collision between the objects as the problem statement suggests, so I will go with the first explanation.
Ad Astra Per Aspera


Force between pair of objects. (none / 0) (#74)
by snowlion on Sat Feb 01, 2003 at 04:14:10 PM EST


"Every object imposing a force on another experiences one of equal magnitude and opposite direction in return from that other object."
Thank God for not pulling out "Equal and Opposite Reaction". Thoroughly confusing.
My preferred way of hearing it is "Forces have a magnitude between two objects, and affects both equally."
That way, you don't have to show that the force that A exerts on B happens to be the same that B exerts on A. Rather, there is only one force existing between them, and thus "two forces" don't need to be lined up in our minds.
I hear that forces actually are carried by tiny messanger particles, and actually DO fly out from objects and impact others. But when we're introduced to classical mechanics, I think it's best to throw that out for the sake of understanding.  Map Your Thoughts



For your version to work, forces would have to be things independent of other objects. They aren't.
Forces are things objects do to one another. Forces cause accelerations. If I apply a force to you, you acclerate. By the third law, you wind up applying a force back on me and I accelerate.
The forces are not independent entities separate from the bodies causing them.
You are right about the messenger particles, though. I plan on saving that for the last chapter, though. I think I'll rework my sentence you quoted too. It looks very clunky when you single it out like that. 8)
Dream Big.
Grow Up. [ Parent ]



My description is that Forces are things that work between objects. It's not difficult to conceptualize that wherever there are two objects, a force is working between them.
"Forces are things objects do to one another." Outside of the particulars (messenger particles), the abstraction works quite nicely. I see no places where you can get into trouble with it, and the benefits of the intellectual simplification are pretty obvious to me.
Instead of saying "I push the wall and the wall pushes back on me," which doesn't go over so well, you can say "The electromagnetic forces repulses you and the wall away from one another."
Seems much clearer to me that way.
"The forces are not independent entities seperate from the bodies causing that." That seems like a pretty metaphysical statement, to me... I'd take a bit of convincing before I accepted that.  Map Your Thoughts [ Parent ]



Most treatments of force define it as the time rate of change of momentum, and momentum is a property of an object, not a pair of objects.
Although it is typical to treat forces pairwise, it isn't necessary, or even done in all cases. Consider, for instance, a classical treatment of a gravitational field and a 5body problem. One way of treating the problem would be to look at it would be to look at the 10 different pairs of equalbutopposite forces, but another would be to compute the gravitational potential field caused by all five bodies, and then work with the five forces caused by the field. Sometimes it's easier to do one approach, othertimes it is easier to do another.
Another examples when working with individual forces as opposed to counteracting pairs is useful is when dealing with pseudoforces in a noninertial reference frame (like a rotating one). What two objects are responsible for Corleosis forces?
Force is dp/dt, and the equalbutopposite rule is a direct consequence of the conservation of linear and angular momentums. That tends to be the way I think of it.
[ Parent ]


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Gun fire is the sound of freedom.


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