Maths - clifford / Geometric Algebra

This page is a copy of a paper by Dr Alfred Differ who has kindly given me permission to reproduce it here. He retains the copyright of this paper.

Introduction to Classical Mechanics using Geometric Algebra (part one)

By adiffer
Sat Jan 25th, 2003 at 05:25:00 AM EST

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 grassmann . 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 æther 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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flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

flag flag flag flag flag flag New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

 

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