Classical Hamiltonian quaternions

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Classical Hamiltonian quaternions were the topic of works written before 1901 on the subject of quaternions. These works are difficult for modern readers because the notation used by early writers, mostly based on the notation and vocabulary of William Rowan Hamilton is different from what is commonly used today. This section provides a description of the original notation, vocabulary and operations used in Lectures on Quaternions and other 19th century books written on the subject of quaternions. For more information on the history of quaternions see the main article on the subject.

1 Informal introduction
- 1.1 Distance squared was viewed as negative time
- 1.2 Quaternion view of space and time
2 Classical elements of a quaternion
3 Operators
4 Cardinal Operations in Detail
- 4.1 Division in Detail
- 4.2 Multiplication in Detail
  - 4.2.1 Distributive
5 Comparison with modern vector notation
6 Notes and references

[edit] Informal introduction

The 19th century classical Quaternion idea was more than just its notation. It was a view of the nature of the relationship between space and time and distance.

[edit] Distance squared was viewed as negative time

In classical quaternion notation a unit of distance squared was equal to a negative unit of time. The Pythagorean theorem, where B = 3i and C = 4j are the sides of a right triangle and A is the hypotenuse would look like:

A 2 = B 2 + C 2

- 25 = - 9 - 16

Classical Quaternion notation used what we today call imaginary numbers to represent distance and real numbers to represent time. To put it in classical quaternion terminology the SQUARE of EVERY VECTOR is a NEGATIVE SCALER^[1] Real number Algebra was called the Science of Pure Time.^[2] A quantity of distance was a different type of number from a quantity of time.

Hamilton objected to calling the squares root of minus one imaginary numbers, saying there is nothing imaginary about them. 19th century writers on the subject of quaternions seldom if ever use the term imaginary. Instead the imaginary part of a quaternion was called the vector of the quaternion. Unlike today in the 19th century complex number was a polite term for imaginary number, but they meant the same thing.^[3]

Classical quaternion notation was controversial because it was introduced with not just one, but an infinite number of square roots of minus one,^[4] and took three of them to use as the bases vectors for a model of three dimensional space as an alternative real Cartesian Coordinates.

[edit] Quaternion view of space and time

Classical 19 century quaternion thinking suggested that real Euclidean 3-space, generally accepted^[5] at the time, might not be the one and only true model of the space and time in which we live.

Quaternion notation on a philosophical level implied that space was of a four dimensional or of a "quaternion" nature, with time being the fourth dimension. This is because if the scaler part of a quaternion was zero that implied that it was a location in space at time zero.

For example if

w + xi + yj + zk = xi + yj + zk

then w = 0

To an extent any model of space and time as a four dimensional entity on a metaphysical level, can be thought of as type of "quaternion" space, even if on a notational and computational level the original classical quaternion four space has continued to evolve.

[edit] Classical elements of a quaternion

[edit] Tensors

19th century texts use the term tensor differently than we do today. What they called a tensor, is what we would today call a unsigned or positive real number. A tensor could shrink or stretch a vector, but could not change its direction.

[edit] Scalar

Scalars are the same today as they were in the 19th century, except that they could be decomposed into a tensor and a plus or minus sign. The operation called take the tensor of, extracted the tensor out of the scalar, resulting in an unsigned real number.

[edit] Vector

In Hamilton's first lecture article 15, he introduces the word vector,^[6] from the Latin vection, or to move. It must be born in mind that Hamilton introduced the concept of a vector.

The meaning of the word vector changed around 1900. Writers in classical quaternion notation used it differently than it is used today. The vector of a quaternion consisted of three what is today called three imaginary units possibly with real coefficients.

In modern terminology a complex number can be understood to mean the sum of two numbers the first being real, and the second being what we today call an imaginary or purely imaginary number. In modern terms a purely imaginary number is one of the square roots of minus one, possibly with a real number coefficient.

The operation of "take the tensor of" decomposed a vector into a tensor and a unit vector. This unit vector had the property that when its square equaled the scalar minus one. This operation would be written s=T(v). Another operation on a vector was get the unit vector. u=U(v).

[edit] Quaternion

The last element classical quaternion notation system was the quaternion which could be represented as the sum of a vector and a scalar.

A quaternion could be decomposed into a scalar and a vector, or into a tensor and a versor.

[edit] Versor

A versor was a special type of quaternion which only affected a vector by rotating it while keeping its length invariant. This rotation could be any number of degrees. The quadrantal versor was a special example of a versor. All quaternions could also be decomposed into the product of a tensor and a versor. Generally a versor consisted of the sum of a scalar and a vector.

[edit] Quadrantal versor

A quadrantal versor has the effect of rotating a vector perpendicular to it by 90 degrees. Hence i × j = k. Here i represents an operator on j rotating it by 90 degrees.^[7] Using i as an operator again i × k = −j. Classical notation viewed this as i operating on k to produce another rotation of 90 degrees. Note the logical consistency here; if it was true that i × (i × j) = −k then it should also be true that (i × i) × j = −k and so i × i must equal minus one.^[8]

In multiplication Minus one was called an inversor, having the effect on any vector of reversing it by 180 degrees to point in the opposite direction. Classical reasoning was that two successive rotations of 90 degrees in the same plane should produce the same effect as one rotation of 180 degrees. Quadrantal versors were therefore called semi-inversors. Quadrantal versors lacked the scalar component of most versors since the scalar component of a versor is the cosine of the angle it rotates a vector about.

[edit] Operators

[edit] Ordinal operators

The two ordinal operations in classical quaternion notation were addition and subtraction or + and -, and they worked pretty much like modern notation.

[edit] Cardinal operations

The two Cardinal operations in classical quaternion notation were x and ÷

[edit] Multiplication

Classical quaternion notation system had only one concept of multiplication. Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation.

Multiplication of a scalar and the vector of a quaternion was accomplished with the same single multiplication operator, multiplication of two vectors of a quaternions used this same operation as did multiplication a quaternion and a vector and the multiplication of two quaternions.

[edit] Division

Classical quaternion notation also had an operation of division. In fact Lectures on Quaternions first introduces the quaternion as the quotient of two vectors, q = a / b. Logically then^[9]^[10] that q × b = a.

[edit] Other important operations

[edit] Taking the scalar and vector of a quaternion

Two important operations in two the classical quaternion notation system were S(q) and V(q) which meant take the scalar part of, and take the imaginary part, what Hamilton called the vector part of the quaternion.

In the classical era this is what the notation looked like:

q = Sq + Vq

Here, q is a quaternion. Sq is the scalar of the quaternion while Vq is the vector of the quaternion.

[edit] Taking the tensor and versor of a quaternion

Another important pair of classical quaternion operations were deconstructing a quaternion into a tensor and versor:^[11]

q=Tq.Uq

[edit] Taking the conjugate

K(q) means to multiply the vector part of a quaternion by minus one.

If q = Sq + Vq then

  Kq=Sq - Vq

[edit] Cardinal Operations in Detail

[edit] Division in Detail

The results of the using the division operator on i,j and k was as follows. ^[12]

$\frac{k}{j}=i$

$\frac{i}{k}=j$

$\frac{j}{i}=k$

$\frac{-k}{i}=j$

$\frac{-i}{j}=k$

$\frac{-j}{k}=i$

$\frac{-k}{-j}=i$

$\frac{j}{-k}=i$

$\frac{-j}{-i}=k$

$\frac{i}{-j}=k$

$\frac{-i}{-k}=j$

$\frac{k}{-i}=j$

[edit] Multiplication in Detail

[edit] Distributive

In the classical notation system, the operation of multiplication was distributive. Understanding this makes it simple to see why the product of two vectors in classical notation produced a quaternion.

q=(ai + bj + ck) x (ei + fj + gk)

q = ae(i x i) + af(i x j) + ag(i x k) + be(j x i) + bf(j x j) + bg(j x k) + ce(k x i) + cf(k x j) + cg(k x k)

Using the quaternion multiplication table we have:

q = ae(-1) + af(+k) + ag(-j) + be(-k) + bf(-1) + bg(+i) + ce(+j) + cf(-i) + cg(-1)

Then collecting terms:

q = -ae - bf - cg + (bg-cf)i + (ce - ag)j + (af-be)k

The first three terms are a scaler.

Letting

w = -ae - bf - cg

x = (bg-cf)

y = (ce - ag)

z = (af-be)

So that the product of two vectors is a quaternion, and can be written in the form:

q = w + xi + yj + zk

[edit] Comparison with modern vector notation

Around the turn of the 19th into the 20th century early text books on modern vector analysis^[13] did much to move standard notation away from that classical quaternion notation, in favor of modern vector notation based on real Euclidean three space.

[edit] The need for i,j,k

Classical quaternions were separated by plus or minus signs, whereas Cartesian coordinates were separated by commas. Hence (w,x,y,z) is different from w + xi + yj + zk where each of the terms i,j,k is a square root of minus one. Cartesian Coordinates represented three space with an ordered triplet of real numbers, (x,y,z). Quaternion notation introduced a different representation.

Vq = xi + yj + zk^[14]

The expression above from 1887 looks a lot like a modern vector, but it is not, it is a point in three space represented using the vector of a quaternion. The i, j and k mean something different than in modern notation.

[edit] Opposition to the square root of minus one

In original Cartesian notation, which existed long before the introduction of quaternion notation, (1,2,3) was very different than (1 + 2 + 3). The first term was a point in three space, but the second one was the number 6. In order to keep up, Cartesian notation needed an i,j,k as well. In this article we use items marked in red to denote modern vector notation as opposed to classical quaternion notation.

But i × i could not equal minus one in the new system, because real Cartesian coordinates consisted of only real numbers.

[edit] Four new multiplications

The classical quaternion notational system had only one kind of multiplication. But in that system the product of a pure vector of the form 0 + xi +yj + kz with another pure vector produced a quaternion.

To add the functionality of classical quaternions to the real three space early modern vector analysis required four different kinds of multiplication.^[15] In addition to regular multiplication which got the name scalar multiplication to distinguish it from the three new kinds, it required two different kinds of vector products. The fourth product in the new system was called the dyad product.

[edit] Dot product notation contrasted with classical quaternion notation

The first new product called · that was computationally equivalent to the classical quaternion operation −S(VA × VB), when applied to vectors of a quaternions A and B was added to the Cartesian model.

But i · i in the new system was +1. And the type of vector in the modern system was different as well; the new vector was not the vector of a quaternion, because it did not consist of a triplet of imaginary components. Rather it was the vector of a quaternion multiplied by the square root of minus one, containing only real components.

[edit] Cross product notation contrasted with classical quaternion notation

The second new product in the new system was the cross product, that was computationally similar to V(VA × VB) or taking the vector part of the product of the vector part of two classical quaternions.

In the new notational system it was still true that (i × j)= k, however, unlike (i · i)=+1 in the second new type of multiplication (i × i)=0. The expressions shown in red here are written in modern notation.

[edit] Dyadic product notation contrasted with classical quaternion notation

The third new product, in the early modern system was the dyad product. It was needed to perform some of the linear vector functions,^[16] that quaternions multiplied into vectors had performed. A dyad was written in some early text books as AB^[17] without a dot or cross in the middle. Three dyads made up a dyadic. This vector product took over the quaternion operations of version and tension. This early aspect of the Gibbs/Wilson system has become more obscure over time.

[edit] New system questioned

In the 19th century supporters of classical quaternion notation and modern vector notation debated over which was best notational system. See history of quaternions for the details.

To provide a vastly oversimplified, short introduction to what motivated these debates consider that in the new notation that i · i =+1, j · j =+1 and k · k =+1. So apparently i,j,k in the modern vector notational system represent three new square roots of positive one.

In the new notational system i, j, and k also apparently represented square roots of zero, since i × i = 0 , j x j = 0, k x k = 0. The new notation system was then based on numbers that were the square root of both zero and positive one. Advocates of the classical quaternion system liked the older idea of a single vector product with a unit vector multiplied by itself being negative one better.

[edit] The quadrantal versor argument

An important argument in favor of classical quaternion notation was that i, j, and k doubled as quadrantal versors. i × (i × j) = −j and (i × i) × j = −j This was not the case in the new notational system of modern vector analysis because their cross product was not associative. In the new notation (i × i) × j = 0, and however i × (i × j) = −j.

Here the text in red is written in modern vector notation.

[edit] Turn of the century triumph of modern vector notation

Modern vector notation eventually replaced the classical concept of the vector of a quaternion.

Advocates of Cartesian coordinates expropriated i, j, and k, along with the term vector into the modern notational system. The new modern vector was different from the vector of a quaternion.

As the computational power of quaternions was incorporated into modern vector notation^[18], classical quaternion notation lost favor. The dyad product and the dyadics it generated, used in early modern works on vector analysis^[19] to perform the linear transform computations done with quaternions also eventually fell out of favor as standard tools of vector analysis as their functionality was replaced by the matrix.

Some early formulations of Maxwell's equations used a quaternion-based notation (Maxwell paired his formulation in 20 equations in 20 variables with a quaternion representation^[20]), but it proved unpopular compared to the modern notation.

The classical vector of a quaternion was multiplied by the square root of minus one and then again by negative one, and installed into modern vector analysis. The computational power of the classical quaternion vector product was exported into the new notation as the new cross and dot products. The computational power of the tensor of a quaternion and the versor of a quaternion became the dyadic, and then the matrix. The scalar and the three vector went their separate ways.

The 3 × 3 matrix rotation matrix took over the functionality of the dyadic which also fell into obscurity.

The scalar-time, 3-vector-space, and rotation matrix-transform had emerged from the classical quaternion and could now march forward as three different mathematical entities, taking with them the functionality of the 19th century classical quaternion. The old notation was left behind as a relic of the Victorian era.

Modern vector and matrix and modern tensor notation had nearly universally replaced Hamilton's quaternion notation in science and real Euclidean three space was the mathematical model of choice in engineering by the mid-20th century.

[edit] Notes and references

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Categories: History of mathematics