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Introduction to Classical Mechanics using Geometric Algebra (part one)

By adiffer in Science
Sat Jan 25, 2003 at 05:25:00 AM EST
Tags: Science (all tags)
Science

This article introduces the reader to the physics theory of Mechanics as it is rendered with geometric algebra. We apply our recently learned skills concerning geometric algebras in kinematics and explore the concepts linking motion to the forces that cause it.

The purpose of this article is entirely educational. If the reader works to comprehend the content and also works the problems, they will come away with a better understanding of Mechanics and how related problems can be solved with geometric algebras.

  1. The reader will understand what is meant by Force and Momentum and how they are defined.
  2. The reader will understand the link between Force and Momentum and the kinematics variables to which they are commonly related.
  3. The reader will understand the basic implications of Newton's Laws and some of the related assumptions that go into them.


Introduction

Mechanics is the first successful theory we count among the subjects that make up modern day Physics; a subject area that was more properly known as Natural Philosophy at its birth. Kinematics helps us to describe motion and provides empirical solutions to real problems. Mechanics takes us up the next step by creating causes for effects that may be used to model motions of any kind. As such, Mechanics gets used in almost all other advanced theories because motions and their causes are literal tools employed by physicists.

The methodology for Mechanics as it was originally described by Isaac Newton in the seventeenth century was based largely upon the mathematical language of geometry. In this language, there were terms to represent numbers and others to describe line, area, and angular magnitudes. The proof that gravity moved the Moon and our cannonballs relied upon the equivalence of the linear magnitudes for how far the Moon would fall toward us in one minute if halted in its forward motion and how far a cannonball falls toward the ground in one second if halted in its forward motion. The fact that our Moon is about sixty times farther from the center of the Earth than we are helped to simplify the calculations.

Mechanics was initially used to handle problems involving the motion of planets, moons, and small bodies through resistive fluids. These problems fall into the modern categories of Celestial and Fluid Mechanics. Today, there are many other variations that employ Mechanics and a new way of distinguishing among them based upon which of our fundamental assumptions is expected to hold and how. The grouping known as Classical Mechanics is the one that holds most faithfully to Newton's original laws and will be the subject of this chapter. The laws and other assumptions will be described shortly.

Classical Mechanics as it is taught today uses a different mathematical language than geometry. In the late nineteenth century, a physicist named J. W. Gibbs wrote a pamphlet for his students describing a simplified vector algebra that was capable of representing Mechanics and the new theory known as Electromagnetism. The system and notation described by Gibbs was a limited form of work done by Grassman. Students first learning Mechanics today are taught using an adapted form of the system proposed by Gibbs. Vectors, dot and cross products, and matrix multiplication are all parts of this language that has expanded Mechanics well beyond Newton's original work.

Whether Mechanics is rendered through expressions using geometry, vectors and vector algebra, or geometric algebra does not change its underlying nature. Translation of that nature from one language to the next is not a radical event. Expecting a new translation to provide greater insight into the order of Nature, however, is somewhat rebellious. Such rebels usually expect everyone else to change to their new approach, after all. Yet this is exactly what the vector algebra provided for Mechanics when it was translated from its original form. It is what we propose geometric algebra will provide again with this translation.

Classical Mechanics is definable relative to Quantum and Relativistic Mechanics by a set of basic assumptions about objects that apply no matter which mathematical language is used to render the theory. These assumptions are listed below. Note that they contain Newton's Laws and a few others we usually assume without writing them down.

  1. Zero Law: Objects have a continuous history in space and time
  2. First Law: Objects have a constant velocity when no forces act upon them as observed from inertial frames of reference.
  3. Second Law: Force causes a proportional acceleration.
  4. Third Law: Every object imposing a force on another experiences one of equal magnitude and opposite direction in return from that other object.
  5. Fourth Law: Multiple forces on one object act as one force that is a sum of all others.
  6. Fifth Law: Two events seen as simultaneous to one inertial observer will be seen as simultaneous to all other inertial observers.
  7. Sixth Law: Physical laws used by one observer will apply for all observers.
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. Our second quantized theories (like QED) come close to breaking this assumption with creation and annihilation of particle/antiparticle pairs, but the physicists step around the issue in a rather deft fashion.
----Technical Note----

If one never asks about what happens to an object before it is created or after it is annihilated, no equations are written that might have to cope with the discontinuity of the existence of the object. The lesson is when a tree falls in a forest with no one around to hear it, do not ask if there is any sound.

The fourth law is one we use as a statement of linearity. If there are two forces acting on one object, the object will accelerate as if one force acted upon it that happens to be the sum of the two real ones. It is an assumption that appears to work in many experiments, but one should never assume it would always work. General Relativity is decidedly non-linear, but it bypasses the issue through the use of curvature to cause accelerations.

The fifth law is the one that distinguishes a relativistic theory from one that is not. Any theory that obeys special relativity breaks this law because as it is written would require the speed of light to vary for observers moving at a variety of speeds relative to a light source.

The sixth law is the one that usually leads to the downfall of most theories while they are at their seed stage. It requires that we only have to solve the puzzles of the universe once for one observer. If our solution is good for one observer, it must be applicable to all other observers because there should be nothing special about any one of us. This is one of the toughest requirements we have for a theory and leads to some of the strangest conclusions including special relativity and some quantum rules from statistical mechanics.

The first, second, and third laws are the main ones students learn when they are taught Mechanics. They deserve a bit of attention, so the remainder of this chapter will focus upon them in some detail.

____________________________________

Section 1: Inertial Reference Frames

Newton's first law states that all objects will experience constant velocity if no forces act upon them and the observer makes the observation from an inertial frame of reference. It is really two statements wrapped into one law, so we must translate both parts.

The first statement defines forces as the cause of a change of velocity. If no forces act upon an object, we may state with confidence that it will not accelerate since its velocity will not change in any fashion. For us, that means the acceleration of an object will be zero for all ranks if no forces act upon it.

Example 1: A spinning top attached to a string at a point along its spin axis.
If this top is spinning around its axis while someone twirls the top around in a circle at the end of the string, both the first and second rank portions of the velocity of the top change with time. This change means a force is present to cause the change. If the top is not spinning around its rotation axis, the rank one piece of the velocity still changes due to the physical movement of the top at the end of the string, so the force causing that acceleration must still have at least one non-vanishing rank.
The second statement is a little more difficult to translate than the first because we must define an inertial frame of reference without resorting to circular logic. In order to do this, we will demonstrate non-inertial reference frames and state that all others are inertial. We will also describe how Newton explained this along with the shortcomings of his approach.
Definition: An inertial reference frame is a special environment in which an observer is not accelerating. The reference frame used by that observer is said to be 'inertial.'
The key to this definition is in knowing when you are not accelerating. This isn't as easy as it sounds. Humans on the surface of the Earth are accelerating as evidenced by the path they take along a latitude circle over the time it takes the Earth to rotate around its equator once. Yet it wasn't long ago when people believed the Earth was fixed and the sky rotated. Sensitive equipment can detect the centripetal acceleration we experience on the rotating Earth if it is moved in latitude, but a zero reading cannot distinguish between a tiny acceleration and no acceleration at all.

In an experimental sense, we really don't know if we are accelerating unless it is obvious that we are. Therefore it is easiest to know when we are not in an inertial frame. Knowing when we are becomes a puzzle requiring a proof of a negative conjecture. In practice, we do the best we can in this regard by looking at our experimental results from many perspectives and by thinking of all the possible accelerations we can that are related to the motion of the observer instead of the motion of the experimental subject.

Newton's approach to this issue was to invent an absolute reference frame that included an absolute clock. The grid of the frame was somehow attached to our underlying universe in such a way that it did not move. This allowed us a way to consider absolute motion relative to it and absolute acceleration through changes to that motion. If such an absolute frame exists, we would be able to discuss whether someone was accelerating or not by describing how the reference frame accelerates relative to the absolute frame. Inertial frames of reference are all those that do not accelerate relative to the absolute one.

To Newton, this was an acceptable solution to the issue. To modern day physicists, it is not acceptable whether we step up to relativity or not. The philosophical problem with this approach is that there is no way to know anything about the absolute reference frame. We can theorize that it exists, but we can't measure it. All our experiments can physically measure are quantities attached to non-absolute frames. Therefore, Newton invented a piece of magic that worked and was not testable with direct probes. He swept this away by stating that objects far from us out among the stars might make good candidates for objects at absolute rest.

----Technical Note----

The issue that cropped up in the late nineteenth century involving the speed of light and detection of it relative to the ćther is actually a different problem. The experiment performed by Michelson and Morely demonstrated a flaw with our conclusion that velocities of objects add like we add forces. This lead to a collapse of the fifth law and users of Mechanics were forced in the direction of special relativity. An interesting note is that much of special relativity was already built into the new theory for electromagnetism.

We will avoid the adoption of an absolute reference frame and accept an observer's best efforts to determine whether or not they are using an inertial reference frame. We will do so in order to avoid inventing a piece of magic and because we know that some day our version of Mechanics must be adapted to conform to the rules of special relativity. Special relativity makes it quite clear that there can be no absolute reference frame.

____________________________________

Section 2: Actions and Reactions

The third law stems largely from our intuitive experience. When one object applies a force to another, the second object applies a force on the first one. The first force is called 'action' while the second one is called 'reaction', but don't get too attached to those terms. Which one gets used depends mostly on the perspective of the observer and we know from law six that such a perspective should not alter the physical laws.

Example 2: Two bowling balls are rolled across the floor toward each other and collide. They bounce away from the collision point in different directions.
A person watching one ball will see it roll along and then suddenly change direction and speed. They know from this experience that the ball was acted upon by a force. A person watching the other ball will come to the same conclusion. An observer watching both balls will notice that the accelerations occur at the same time, so they might suspect that the forces have something in common.
Law three encodes the expectation from the example above that the presence of one force is matched by another where the terms 'projectile' and 'target' are swapped. In practice, this law works quite well when one tracks only the first rank portion of acceleration. It remains to be demonstrated that it works well for other ranks at the same time.

There are actually two versions of Newton's third law that are referred to as the strong and weak forms. The strong form requires the reaction force to be of equal magnitude and opposite direction compared to the action force. The weak form drops the requirement for the direction. Because we expect our forces to be multi-ranked in general, the definition of direction becomes problematic. Individual ranks can be said to have direction, but their sum might not unless all ranks but one vanish. Since this will not be the case in general, we will hold to the weak form of the third law with the added expectation that reaction forces be opposite in direction relative to their action equivalents on a ranks by rank basis. This translated version of the third law will do for now. Experiment will determine its validity.

----Technical Note----

It is quite possible some operator will do the job and deliver a strong form of the third law for us. Reversion will flip some ranks and not others. Parity is similar but with different outputs.

____________________________________

Section 3: Force and Momentum

Newton's second law is where the real action takes place. It is the law that links causes (forces) to effects (accelerations.) It is the step that takes us beyond our empirical science of kinematics by explaining what causes the motions we observe. It does not explain why forces exist or how they might act, but it does postulate a relatively simple motivator of motion that can be applied to a variety of problems in a framework that is quite testable.

From the first law, we know that if no forces act on an object the object will not accelerate. Because the relationship between acceleration and velocity is one of rate of change, we will create a concept known as momentum that is similarly related to force. If a force exists, momentum is changing. Mathematically speaking we write it as follows

F = δP/δt

where P is momentum, δP is the change to the momentum, and δt is the time elapsed for that change.

----Technical Note----

Newton referred to momentum in his Principia as 'motion.' He defined it as a product of the velocity and quantity of matter of an object.

We invent the concept of momentum in order to create some property an object has that changes when forces occur. We know from law one that velocities change when forces exist, but velocity is something we measure about an object relative to other objects. Velocity is not really a property of an individual object unless one adopts an absolute reference frame. Momentum fills that role a little better by being associated directly with a single object. It is actually a combination object as will be shown later. Note that the absolute value of momentum does not really matter yet since forces are related to the change of momentum. Only δP matters.

So far we have invented a notion called force that must exist when velocities change. We have added another invention named momentum that acts as a wellspring for forces since we required it to be the thing about an object that really changes when a force exists on the object. We will invent other notions later, but these two will do for now.

Example 3: Roll two bowling balls and collide them again.
Each ball has a certain amount of this mysterious property called momentum. After the collision, each one has a different amount. We conclude that forces must have existed on each in the amount δP1/δt for the first and δP2/δt for the second. Because forces existed, the balls accelerated during the collision, hence, changed their velocities.

In theoretical discussions, this chain of logic is run forward from our starting point with momentum. In experimental observations, the chain is run backwards since we notice the velocity change.

Newton's second law defines a link between force and acceleration that should allow us to calculate one from the other. This link, as a result, also connects momentum and velocity. The mathematical form for this link is not as obvious for us as it was for Newton, though. We know a simple scalar can be used to link the first rank portions of force and momentum to acceleration and velocity respectively, but that scalar won't work as well for second rank pieces without a fix to kinematics we could have introduced earlier in a confusing and non-motivating manner.

To the kinematics variables for location, velocity, and acceleration, we add our new ones for Mechanics named momentum and force. Because the kinematics variables are multi-ranked, we expect the new ones are too. However, in our effort to link them, we must face a complexity we swept under the rug in the last chapter. This complexity is the one concerning the units we use for our variables. Resolving the unit issue removes an annoying thorn from our side and happens to fix an apparently unrelated issue we will describe shortly.

----Unit Fix----

Consider the location L of the fly in our room we described in the last chapter. We assigned the position, orientation, and volume data for the fly to the first, second, and third rank portions of L respectively. We tacked on the value of a clock using the scalar portion too. This location object contains four apparently different things within it and each uses different units.

The position of the fly is measured in units of length by noting how many pin lengths are required along each of the three reference line segments to reach the fly. The volume of the fly is measured in units of length cubed by noting how much of the block defined by the pins is needed to occupy the same space as the fly. The orientation of the fly, however, does not use area units. It uses angles. As such, it is the odd one out.

Yet we could have used area units for orientation information if we had imagined the angles as pie slices of a unit circle. The value of an angle is equivalent to the area of a circular wedge described on a unit circle. Had we done this our location components would have had units of second, meters, square meters, and cubic meters for the scalar, line, area, and volume segments respectively.

An even better approach for orientation information would have been to use a representative area for the fly and project it onto the reference plane segments to get the three pieces of data we need to know which way the fly is facing. The representative area used could be any cross section of the fly and it would work fine. As long as we do not choose a different cross section later, the information we glean from the projections is equivalent to orientation information.

The last unit that seems out of place is the 'second' used for the scalar value of time. We will side step this one by noting that Kinematics makes no requirement that time have any units at all. All we needed for time was a label we could use to impose a sense of order on sets of locations. The fact that some people like to consider a unit named 'second' for this label should not bother us or force us to treat it as a unit on the same level of importance as the meter. So we will describe the units for location as follows.

Location units are ( meters0, meters1, meters2, meters3) or ( unitless, m, m2, m3) or ( second, m, m squared, m cubed) with seconds being effectively unitless much as radians are.
----Cause and Effect Link----

Through out this and future chapters, orientations will be determined in this new way for the sake of logical and unit consistency. The issue was saved for this chapter instead of being resolved in the last chapter because there is no clear motivation to choose one way or another, let alone to demote the unit of time to something unitless, until we are faced with Newton's second law. With this choice made, the link between force and acceleration falls out as a simple case of scalar multiplication by something we refer to as inertia. (Newton referred to it as 'quantity of matter.')

Force is proportional to Acceleration. The proportionality constant is called inertia.
The relationship between momentum and velocity falls out as naturally.
Momentum is proportional to Velocity. The proportionality constant is called inertia.
----Technical Note----

It is mathematically reasonable to add an additional multi-ranked constant to the equation for momentum and velocity, but we will avoid doing it here since it would be equivalent to a measurement of the momentum of the entire universe. If the whole universe were speeding off in some direction and rotating and expanding, the base momentum for objects within it wouldn't be zero. The base would be that constant. Since forces occur when momentum changes, though, we really only care about δP and not the absolute value of P. This won't change unless someone thinks up a way to measure the absolute values.

The scalar constant called inertia is the link between cause and effect in the second law. It wouldn't have been good enough, though, if we had not fixed our unit inconsistencies. Scaling rotational acceleration by inertia isn't enough to get torque since the size of the object with the inertia matters. Use of a cross sectional area through the object fixes this, though, by giving the rotational acceleration units of area per second per second. The size is built into the relationship if the proper units are used. This potential issue is the one we fixed when we made our choice regarding units.

----Technical Note----

Note that the equation of motion has four parts to it since there are four possible ranks within force and acceleration. The rank one piece is the one traditionally referred to as Newton's second law.

The reader who has managed to keep up thus far will have noticed the relationship between the kinematics variables L, V, and A is mostly reproduced with P and F. All we are missing is an equivalent for L. There is no harm in inventing one by defining it as the thing that changes to cause momentum. Whether it is useful in physical theory remains to be seen, though. We will call it M.

To see that these inventions break down to Newton's original laws and other things we already know from physics, we must break out the various ranks into separate statements and link them back to their counterparts in traditional Mechanics.

Example 4: Rank zero (scalar) portion of the equations of motion
We know from the last chapter that the scalar part of L is used to represent our time label. We also know that V is δL/δt. So the scalar portion of V is constant and the scalar part of A vanishes. Therefore the scalar parts of P and F are constant and zero respectively.
Example 5: Rank one (vector or line segment) portion of the equations of motion
The rank one segments of L, V, and A are the parts included in traditional Kinematics. The rank one parts of P and F are counterparts for the traditional linear momentum and linear force vectors. Singling out this rank gives us the following.
Linear Force = inertia * linear acceleration
Linear Momentum = inertia * linear velocity
Linear M = inertia * position = (an inertia weighted position).
The first two are exactly what is taught for the traditional approach to Mechanics. The third equation might be useful later to help determine something known as the center of mass for a system of objects.
Example 6: Rank two (bivector or plane segment) portion of the equations of motion
Rank two portions of L, V, and A are the counterparts to the traditional kinematics variables, but with a change of units to convert from angles to areas. Angular velocities and accelerations become area velocities and accelerations. Singling out this rank, then, gives us the following.
Area Force = inertia * Area acceleration = inertia * |representative area| * angular acceleration
Area Momentum = inertia * Area velocity = inertia * |representative area| * angular velocity
Area M = inertia * |representative area| * orientation.
Another way to look at the first two connects them better to traditional equations. Let rotational inertia be the inertia multiplied by the representative area and we get the following.
Area Force = rotational inertia * angular acceleration
Area Momentum = rotational inertia * angular velocity
From these equations, we can identify the second rank parts of P and F as the dual of the angular momentum and torque respectively.
Area Force = dual(torque)
Area Momentum = dual(rotational momentum)
Don't be fazed by the dual operation. This is present only because of the convoluted definition of the cross product used by people who learned to use the vector algebra.
Example 7: Rank three (trivector or volume segment) portion of the equations of motion
Rank three parts of L, V, and A do not have traditional counterparts in Kinematics, but we assigned them to hold volume information about an object. More work must be done later to discover the part of traditional physics that is probably encoded in this rank of the equations of motion.
----The Technique----

We wrap up this section by describing the technique shown above in an abstracted manner. We started with the kinematics variables L, V, and A with which we describe externally observable things about objects. To measure them we set up a good reference frame and clock and project out the information we need from a frame attached to the object. None of the variables explain how the observed information comes to be what it is, though.

In order to explain motions, we invent another set of variables similar to the kinematics ones and name them M, P, and F. These are to be the causes we need to answer questions that start with the words 'how' and 'why.' To link them to L, V, and A we postulate the simplest connection we know by making them proportional. The proportionality constant is named inertia or mass depending on how precise we wish to be. This is the postulate that is encoded in Newton's second law, so nothing revolutionary is occurring.

The inertia is supposed to be a property inherent to an object. No external reference frame needs to be set up to know it exists. It is what it is for every object in our universe. This makes M, P, and F a combination of externally measured variables and properties inherent to a body. This is how they can act as causes to the kinematics variables.

The product of properties inherent to a body and the external kinematics variables is going to appear all through Mechanics and related subjects in later chapters. Later situations will deal with properties that are not as easily represented as the inertia, though. Those future properties may need higher ranked representations above scalar, but the general technique we have developed here will remain intact. Causes will be products of inherent properties and kinematics variables.

____________________________________

Summary

In this section we developed the basic assumptions and laws for a theory of Mechanics based upon geometric algebras. The basic expectations for the scope of a Mechanics theory were discussed along with the distinctions that separate classical, quantum, and relativistic Mechanics. We finished with a focus on how Newton's three laws translate and what the implications are for us regarding units and the multi-ranked nature of our variables.

We showed that Newtonian Mechanics is largely unaltered when translated in the mathematical sense to use geometric algebras. Traditionally disparate quantities are brought together as single ranks of multi-ranked, umbrella-like objects, but no alterations to the observable physics would result. Forces are still causes of accelerations and momentum is still the thing that must change for a force to exist. Even the inertia got to keep its typical mystique as a proportionality constant of scalar rank, though we did have to adapt to a different type of orientation unit than we had originally planned to use from Kinematics.

In the next section, we develop Mechanics a bit further by studying a few special cases with simple or symmetric forces. From these cases we will define other concepts typically included in Mechanics texts and useful for solving certain classes of problems.

----Problems for Mechanics (part one)----

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?

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?

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.

4: Suppose you observe a large ball placed in the middle of a large frictionless floor. You note it isn't moving. You check back later and notice it is moving slowly to one side of the room with a backspin. If a single, linear force were applied to the ball, would it move this way? Describe the nature of the force that leads to the observed momentum and how it can be represented.

5: Consider example seven again. What type of force can lead to a change in volume for an object? What units should it have? What would the 'volume momentum' be then?

____________________________________

References:

----Traditional Mechanics Sources----

  1. Philosophiae Naturalis Principia Mathematica
  2. A Lecture Note Set regarding Mechanics U Texas @ Austin
  3. Harvard's Physics 11A
  4. U Mass' Physics 421
  5. Audio Lectures: Hold onto your hat.

----Academic Groups/Societies/People----

  1. Clifford Algebra Society
  2. The Geometric Algebra Research Group: Cavendish Laboratory
  3. Geometric Calculus R&D: ASU
  4. Leo Dorst's page

----Self References----

  1. Representing objects and properties in Physics
  2. Usage of Geometric Algebra as a representational tool
  3. Introduction to Geometric Algebra: part one
  4. Introduction to Geometric Algebra: part two
  5. Introduction to Geometric Algebra: part three
  6. A Description of Motion with Geometric Algebra

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Related Links
o Isaac Newton
o Classical Mechanics
o J. W. Gibbs
o Grassman
o Relativist ic Mechanics
o QED
o General Relativity
o experiment
o Michelson
o Morely
o Principia
o inertia
o Philosophi ae Naturalis Principia Mathematica
o A Lecture Note Set regarding Mechanics
o Harvard's Physics 11A
o U Mass' Physics 421
o Audio Lectures: Hold onto your hat.
o Clifford Algebra Society
o The Geometric Algebra Research Group: Cavendish Laboratory
o Geometric Calculus R&D: ASU
o Leo Dorst's page
o Representi ng objects and properties in Physics
o Usage of Geometric Algebra as a representational tool
o Introducti on to Geometric Algebra: part one
o Introducti on to Geometric Algebra: part two
o Introducti on to Geometric Algebra: part three
o A Description of Motion with Geometric Algebra
o Also by adiffer


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Introduction to Classical Mechanics using Geometric Algebra (part one) | 78 comments (55 topical, 23 editorial, 0 hidden)
You forgot to mention (3.00 / 4) (#17)
by Noam Chompsky on Fri Jan 24, 2003 at 09:27:57 PM EST

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,

Does not (none / 0) (#32)
by danni on Sat Jan 25, 2003 at 11:12:06 AM EST

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 ]

I'm afraid not. (none / 0) (#34)
by Noam Chompsky on Sat Jan 25, 2003 at 12:22:18 PM EST

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 ]

That's what makes them "paradoxes." (none / 0) (#35)
by Noam Chompsky on Sat Jan 25, 2003 at 12:31:33 PM EST


--
Faster, liberalists, Parent ]

+1 FP but... (3.00 / 1) (#21)
by yicky yacky on Fri Jan 24, 2003 at 10:26:15 PM EST

...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
Sure (5.00 / 1) (#23)
by adiffer on Fri Jan 24, 2003 at 11:42:45 PM EST

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 ]

Unnecessary. (none / 0) (#26)
by yicky yacky on Sat Jan 25, 2003 at 12:06:36 AM EST

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... Op-Eds (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 ]
Please do this! (none / 0) (#28)
by tzigane on Sat Jan 25, 2003 at 01:16:12 AM EST

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 ]

My cat (none / 0) (#29)
by tzigane on Sat Jan 25, 2003 at 01:22:21 AM EST

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 ]

Hmmm yes. (none / 0) (#36)
by yicky yacky on Sat Jan 25, 2003 at 02:18:00 PM EST

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 part-time 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 semi-stationary eye-in-the-sky 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 ]
A general principle: (1.50 / 2) (#31)
by Estanislao Martínez on Sat Jan 25, 2003 at 03:46:10 AM EST

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

--em

I agree (none / 0) (#38)
by adiffer on Sat Jan 25, 2003 at 03:11:31 PM EST

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 ]

So... (none / 0) (#42)
by Estanislao Martínez on Sat Jan 25, 2003 at 09:58:06 PM EST

Two questions.
  1. 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?
  2. 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 ]

Answers from a third party (none / 0) (#44)
by PhysBrain on Sun Jan 26, 2003 at 04:03:59 AM EST

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 ]

countless (none / 0) (#45)
by adiffer on Sun Jan 26, 2003 at 04:10:00 AM EST

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 ]

Some suggested reading (none / 0) (#47)
by PhysBrain on Sun Jan 26, 2003 at 05:01:16 AM EST

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 lay-person.  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 ]

You have forgotten one thing. (3.00 / 1) (#33)
by Krapangor on Sat Jan 25, 2003 at 12:11:25 PM EST

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 ]

Domains of accuracy (none / 0) (#39)
by adiffer on Sat Jan 25, 2003 at 03:24:35 PM EST

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 meta-knowledge for this branch of Physics is deep and rich.  Anyone wanting to improve their skills as a theory maker must get this kind of meta-knowledge 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 meta-knowledge, 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 ]

thanks (none / 0) (#48)
by adiffer on Sun Jan 26, 2003 at 05:12:20 AM EST

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 multi-ranked, but that doesn't really change the meta-rules 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.
Brain Dropping (3.00 / 2) (#41)
by anyonymous [35789] on Sat Jan 25, 2003 at 09:28:47 PM EST

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.

Quantum Zero Effect (5.00 / 1) (#43)
by fatllama on Sun Jan 26, 2003 at 02:31:06 AM EST

The effect is studied and is known as the Quantum Zeno Effect. It's basically a reassertion of the A-Watched-Pot-Never-Boils 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 Anti-Zeno 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 ]

I think you're right (none / 0) (#56)
by anyonymous [35789] on Sun Jan 26, 2003 at 11:03:03 PM EST

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 ]

chomp chomp gulp (5.00 / 1) (#61)
by adiffer on Mon Jan 27, 2003 at 10:05:28 PM EST

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 question... (3.00 / 1) (#49)
by johwsun on Sun Jan 26, 2003 at 07:18:23 AM EST

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?

I mean .... (none / 0) (#50)
by johwsun on Sun Jan 26, 2003 at 07:26:45 AM EST

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 ]

smoothness (none / 0) (#51)
by adiffer on Sun Jan 26, 2003 at 12:04:37 PM EST

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 ]

Continuity (none / 0) (#52)
by PhysBrain on Sun Jan 26, 2003 at 12:21:27 PM EST

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 space-time 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 ]

labels and axes (none / 0) (#53)
by adiffer on Sun Jan 26, 2003 at 06:29:12 PM EST

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 ]

A rough guess (none / 0) (#57)
by PhysBrain on Mon Jan 27, 2003 at 04:52:19 AM EST

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 multi-ranked 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 self-consistent model, then I will await your further posts with much interest.

Ad Astra Per Aspera
[ Parent ]

current guess (5.00 / 1) (#62)
by adiffer on Mon Jan 27, 2003 at 10:10:26 PM EST

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 ]

Define 'Rank' (none / 0) (#54)
by MyrddinE on Sun Jan 26, 2003 at 08:58:19 PM EST

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 pre-requisites 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 == Grade (none / 0) (#58)
by PhysBrain on Mon Jan 27, 2003 at 05:06:57 AM EST

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 ]

Not standalone... (none / 0) (#59)
by BlaisePascal on Mon Jan 27, 2003 at 05:09:07 AM EST

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 "space-like" coordinates and no "time-like" coordinates.

R(3,0) is defined in relation to three different ortho-normal basis vectors (called e1, e2, and e3 for convenience), and a real-valued 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, non-commutative, so the order of multiplications needs to be correct, and the following simple simplification rules:  eaeb can be writen as eab, eaa = e, and eab = -eba, 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 non-equal a, b, eabeab = eabab = -eabba = -eaa = -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 (e123). 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 vice-versa, in a one-to-one 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

e123 e123 = e123123

=e112323

=e2323

=-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 ]

Discussion of problem one (none / 0) (#60)
by PhysBrain on Mon Jan 27, 2003 at 05:38:15 AM EST

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

yup (none / 0) (#63)
by adiffer on Mon Jan 27, 2003 at 10:20:56 PM EST

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 ]

Energy.... (none / 0) (#64)
by BlaisePascal on Tue Jan 28, 2003 at 12:35:35 AM EST

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?

right track (none / 0) (#65)
by adiffer on Tue Jan 28, 2003 at 02:44:21 AM EST

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 ]

problem five idea (none / 0) (#66)
by adiffer on Tue Jan 28, 2003 at 03:08:33 AM EST

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 meter3/second2.

If you take one of these forces and apply it across an area (divide by m2) you should get the linear acceleration of that surface, right?

-Dream Big.
--Grow Up.

Pressure (none / 0) (#68)
by PhysBrain on Tue Jan 28, 2003 at 06:02:05 AM EST

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 time-dependent 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 Navier-Stokes 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 ]

Sounds good (none / 0) (#69)
by adiffer on Tue Jan 28, 2003 at 04:09:52 PM EST

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 ]

GA library (none / 0) (#70)
by PhysBrain on Tue Jan 28, 2003 at 05:21:16 PM EST

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 Vn.  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 ]

PlanetMath (none / 0) (#73)
by adiffer on Thu Jan 30, 2003 at 03:05:46 PM EST

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 ]

Discussion of problem two (none / 0) (#71)
by PhysBrain on Tue Jan 28, 2003 at 06:51:11 PM EST

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: non-inertial 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

Discussion of problem three (none / 0) (#72)
by PhysBrain on Wed Jan 29, 2003 at 11:06:20 PM EST

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

hmm (none / 0) (#75)
by adiffer on Sat Feb 01, 2003 at 04:54:28 PM EST

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 ]

Forces BETWEEN Objects. (none / 0) (#76)
by snowlion on Sat Feb 01, 2003 at 07:47:21 PM EST

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 ]

Forces and momentum. (none / 0) (#77)
by BlaisePascal on Mon Feb 03, 2003 at 04:13:50 PM EST

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 5-body problem.  One way of treating the problem would be to look at it would be to look at the 10 different pairs of equal-but-opposite 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 non-inertial reference frame (like a rotating one).  What two objects are responsible for Corleosis forces?

Force is dp/dt, and the equal-but-opposite rule is a direct consequence of the conservation of linear and angular momentums.  That tends to be the way I think of it.

[ Parent ]

My cat's name is mittens (1.66 / 3) (#78)
by QuantumG on Fri Feb 07, 2003 at 12:04:58 AM EST

her breath smells like cat food.

Gun fire is the sound of freedom.
Introduction to Classical Mechanics using Geometric Algebra (part one) | 78 comments (55 topical, 23 editorial, 0 hidden)
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