Classical Hamiltonian quaternions

Classical Hamiltonian quaternions were the topic of works written before 1901 on the subject of quaternions. Quaternion Calculus was a complete system of mathematics that included its own notion of scalars, vectors and quaternions and how these entities were interrelated. These works can be difficult at first 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,Elements of 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.

Informal introduction

The 19th century classical Quaternion idea was more than just its notation. For Hamilton and those devoted to his school including Tait it was a view of the nature of the relationship between space and time and distance.

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:

$A^2\; =\; B^2\; +\; C^2$

$-25\; =\; -9\; -\; 16$

To put it in classical quaternion terminology the SQUARE of EVERY VECTOR is a NEGATIVE SCALER Scaler Algebra was called the Science of Pure Time.

Since in classical thinking a quantity of distance had a negative square it was a different type of number from a quantity of time, because time like numbers were represented by numbers having a positive square.

Classical quaternion notation introduced with not just one, but an infinite number of square roots of minus one, and took three of them to use as the orthogonal bases vectors for a model of three dimensional space, that was intimately connected to a fourth time dimension, as an alternative three dimensional Cartesian Coordinates.

Quaternion view of space and time

Classical 19 century quaternion thinking suggested that real Euclidean 3-space, generally accepted 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, consisting of time closely linked with three spacial dimensions. This is because if the scalar 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 (one time plus three directional space) has continued to evolve.

Contexts of Quaternion Geometry and Linear Algebra

A source of great confusion is that quaternion geometry and multi linear algebra are two completely different contexts. Multi linear algebra and classical quaternion nomenclature share many of the same words, and both can be used to study the geometry of space time. But the same word can often mean two very different things in the two different contexts.

Three important examples of words that have both a linear algebra context and a classical quaternion geometry context are vector, tensor and space . Algebra and even the word quaternion have taken on different meanings in these two different branches of science.

Linear algebra exclusive point of view

Contrary to the classical quaternion point of is the view that some extension of linear algebra is best suited for all problems both systems attempt to address. This point of view seeks to extend algebra by extracting ideas from the old classical quaternion system and then proclaim that classical quaternion geometry is musty, obsolete and unnecessary, and has been replaced by some new extension of algebra. This view existed in both the 19th and 20th century, and still exists in the 21st.

A factor related to this view that makes it difficult for some beginners to understand commentary written from a linear algebra exclusive point of view on classical quaternion thinking is the use of Quaternion negative nomenclatures. Modern quaternion negative nomenclatures tend to use words with negative connotations for classical quaternion concepts. For an example of the use of quaternion negative terms consider the phrase 'a type of complex number' in reference to a versor.

On the other hand quaternion negative nomenclatures tend to use positive terms for concepts developed in the context of linear algebra. One of the most glaring examples of the built in bias of these quaternion negative nomenclatures is calling a entity consisting exclusively of three-tuples of numbers with a time like quality who have a square of plus one real, and numbers with a distance like quality that have a square of minus one such as classical vectors imaginary.

Classic Isomorphism Tests

Isomorphic means of the same shape. Extensions to linear algebra often claim to have discovered a new system that is 'isomorphic' to the quaternions. These claims can be confusing because they often involve first redefining the classical quaternion and then introducing the new entity that is actually isomorphic to this redefined quaternion.

=The Tait Test=

The Tait test is to count the number of kinds of multiplication in a system claiming to be isomorphic to the quaternions. If there is more than one cardinal type of multiplication from the point of view of the important classical thinkers of the later period lead by Peter Guthrie Tait the entity is not a classical quaternion. 19th century debate raged over this issue with regard to early versions of the notion of a normed vector space over the reals lead by Gibbs, Heavyside and Wilson, who were following in the venerable footsteps of Euclied, Descartes and Newton.

=The Fronobious Test=

A second test is the division test. If the extension to linear algebra claiming to be a quaternion does not define division then it is not in the classical view isomorphic to the classical quaternion.

The next question to ask is if this new entity is closed under division. This can be problematic for matrix representations that claim to be isomorphic to the classical quaternions. Finding a way to start with two alleged classical quaternions and using the operations of that system arrive at a matrix that does not have an inverse then from the strict classical point of view you have proven that since the system is not closed under division, it is not isomorphic to the classical quaternions.

With these tests in mind a student of the classical view of quaternions can avoid the confusion that results from reading modern commentaries that use the word quaternion in a different context from the context meant in the classical texts.

Classical era nomenclature objection

Hamilton objected to calling the square roots 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 what from a linear algebra point of view is considered the imaginary part of a quaternion was in the quaternion context called the vector of the quaternion.

Expropriation

Quaternion negative nomenclatures often expropriate terms from classical quaternion nomenclature and redefine them to a point that it makes very difficult to understand classical texts. The i,j,k of linear algebra and the so called scalar product or dot product and vector product or cross product are examples of expropriated ideas. In classical quaternion thinking there is only one product.

Hence although Hamilton coined the phrase vector, and pioneered the concept of four dimensional space and time, quaternion negative nomenclature can construct sentences like "the vector of a quaternion is not a real vector". In the context of classical quaternion thinking this is basically gibberish. It reflects a linear algebra exclusive point of view.

Some modern thinking uses the word quaternion to refer to some extension of modern algebra that has little in common with the classical quaternion. This makes it hard not only to distinguish between their new extension of algebra and the classical quaternion, but between the extensions themselves.

Vectors

See also main article Vector of a quaternion

See also section of this article on the classical vector as an element of a quaternion.

Writers in classical quaternion notation used the word vector differently than it was used by the rival field of vector analysis.

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 vector part of a quaternion consisted of three what is today called three orthogonal imaginary units possibly with real coefficients.

The use of the word vector, meaning vector in the classical quaternion context began a steep decline around 1900.

Tensors in linear algebra and classical quaternion theory

The Hamiltonian space made up of quaternions is non-Euclidean.

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 positive real number.

In multi-linear algebra tensor means a multi-dimensional array of numbers. The tensor of a quaternion is a single number not an array. What is meant by a tensor now is generally what would be written in quaternion notation as multiple acts of tension, using the product of several quaternions to stretch an object in several directions.. The tensor of a quaternion is a tensor of order zero. When a tensor is multiplied with a vector it has the effect of making it longer or shorter but can never change its direction. The tensor of a vector in quaternion theory has a lot in common with the Euclidean norm. The tensor of a quaternion has the same formula as the R4 norm. But the tensor of a quaternion plays a very different role in H than the norm of R4.

Recalling Euler theorem that a in order to rotate an object from any one orientation to any other orientation requires three angles. Hence a general transform of a vector in space would require three quaternions.

$s(r(qBq^\{-1\})r^\{-1\})s^\{-1\}$

Likewise an entity in general can be subjected to acts of tension in three different directions, by a triple operation. That is assuming that B is a vector in this expression.

$(s+r+q)B$

With proper choice of basis versors the Tensors of these three quaternions Ts, Tr, Tq have some degree of correspond to the modern notion principle axis of the modern stress tensor.Early works on vector analysis called a conglomeration of the somewhat quaternion like entity called a dyad a 'right tensor', and describe a different notion of breaking an operation down into a tensor and a versor. However the versor is closer to a dyadic implementation of a three by three matrix as is stated in the source.

A general transformation of four space would required four quaternions. Beware that unlike Einstein notation, quaternion notation does define powers of quaternions.

$g(s(r(qBq^\{-1\})r^\{-1\})s^\{-1\})g^\{-1\}$

Since a quaternion always consists of one time dimension and three space dimensions the notion of using subscripts and superscripts to denote covariance and contravariance is somewhat redundant.

Quaternion exclusive point of view

In the classical period some thinkers were guilty of taking a quaternion exclusive point of view.

For example, in the classical quaternion notation system there is only one kind of multiplication or on other words only one kind of vector product.

In the latter part of the classical period advocates of an exclusive quaternion view took advantage of this and used exclusion tactics in reverse. When Gibbs first attempted to expropriate and redefine the word vector, and the symbols i,j and k from the classical quaternion system Tait called the new system hermaphraditical to express his quaternion positive algebra negative bias, in reference to the fact that in Tait's view the new entities apparently had two sets of reproductive organs and belonged in a freak show. Tate was intentionally using a Quaternion Positive, Algebra negative nomenclature by assigning a word with at the time a very negative connotation to Gibbs concept of a vector.

Tait was fond of using the term real distances, for numbers with the square root of minus one. His motive was to use nomenclature that gave his system more appeal.

An example of a quaternion positive nomenclature would be to call the versor of a quaternion written Uq an uncomplicated number.

Frobenius theorem

The 1877 Frobenius theorem is very helpful.

From an algebra context proved that real numbers, complex numbers and quaternions were 'the only possible division rings over the reals'.

From a classical quaternion point of view it draws a clear dividing line between quaternion math and linear algebra by proving what is possible and impossible.

In quaternion math there are three possible closed spaces. A space that exists of just one dimension time or in the context of algebra $R^1$. A space that consists of time dimension and one distance like dimension $C1$. A space that consists of three distance like dimensions and one space like dimension. All these spaces are subsets of a quaternion.

Hamilton states early on that any reasonable algebra should include the operations of addition, subtraction, multiplication and division. Hamilton died in 1865, before Frobenius theorem.

Impossible yet useful

A quaternion positive algebra positive point of view sees both algebra and quaternions as useful. Few if any took this view in the 19 century.

An important variant on this last point of view is the quaternion positive algebra positive but impossible view. This view did not exist in the 19th century, but it goes like this:

Just because a space is physically impossible and does not exist, does not imply that it is not useful. In order to solve a system of linear equations in five unknowns the notion of an impossible space consisting of five time dimensions is very useful indeed. Just because Newtonian mechanics is physically impossible because it does not work if you go to fast does not mean that it is not very useful.

Nobody in the 19th century had ever seen anything solid go fast. When in the 19th century Frobenius proved that Euclidean space was impossible, and when in the 19th century Maxwell used the math that Hamilton invented and figured out that light always traveled at the speed of light, which also proved that Euclidean space was impossible, the meaning of these things was incomprehensible.

To the 19 century mind, Euclidean space and the Newtonian physics based on it were both useful and possible. More than possible 19 century thinkers thought Euclidean space was something real. They even called it real space. Real space is an idea so useful that the idea it was also impossible was inconceivable to the 19th century mind.

Classical elements of a quaternion

Tensors as a Type of Number

The tensor of a quaternion played an important role in classical quaternion theory.

A tensor is a very time like quantity. It can only go forward. A tensor does not need a positive or negative sign.

When you multiply a tensor and a vector you get a new vector which is longer or shorter, but still has the same direction. This is called performing an act of tension. Tait called a tensors the stretching factor

Because tensor could shrink or stretch a vector, but could not change its direction. A tensor could approach zero as a limit, but zero is not a tensor.

The product of two tensors is another tensor, the sum of two tensors is another tensor, and the quotient of two other tensors is another tensor.

However when two tensors are subtracted from each other the result can possibly be a new type of number.

For example x = 3 - 5 has no solution within the family of numbers called tensors. -2 is an example of a new and different kind of number called a scalar.

The symbol T stands for the operation called 'take the tensor of' and it is used to take the tensor part out of any of the other kinds of numbers that exist in the classical quaternion system.

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.

Vector

See main article The vector of a quaternion

Every quaternion can be decomposed into a scalar and a vector.

$q\; =\; S(q)\; +\; V(q)$

These two operations S and V were called take the Scalar of and take the vector of a quaternion. Vq and Sq could be written with out ambiguity.

The operations of "take the tensor of" and take "the versor of" could then decomposed the vector of a quaternion V(q) further into a tensor and a unit vector. Like all vectors this unit vector had the property that its square equaled the scalar minus one.

The first of these operations would be written s=T(v). The second operation, taking the versor of a vector would return unit vector. u=U(v). A unit vector is also a special type of versor with an angle of 90 degrees, hence a unit vector can rightfully be called a special type of versor called a right versor.

Vq = (T.Vq)(U.Vq)

Hamilton introduced the world to the concept of a vector in the 1840's. In Hamilton's first lecture article 15, he introduces the word vector, from the Latin vection, or to move.

Versor as a type of number

The versor of a quaternion is a special type of quaternion with useful properties.

The tensor of a versor is always equal to one.

In general a Versor can be associated with a plane, an axis and an angle.

A versor can also in general be represented by a unique great circle arc. This arc is greater than zero and less than 180 degrees. This is because the shortest distance between any two points of a sphere has a maximum limit of an arc corresponding to 180 degrees.

Right versor

When the arc of a versor has the magnitude of a right angle, then it is called a right versor or quadrantal versor.

Like all quaternions a versor can be decomposed into the product of its tensor and its versor.

The versor of a versor is the same as the versor UUq = Uq

As with other quaternions, a versor consists of the sum of a scalar and a vector.

Radial Quotient

The ratio of two vectors of equal length is called a radial quotient or a radial.A versor may also be viewed as the quotient of two vectors which are equal in length. In this case the arc can be visualized as the arc connecting the two vectors when they are placed tail to tail. In this representation the plan of the versor is the plane of the two vectors and the axis of the versor is a unit vector perpendicular to the plane.

Degenerate forms

The scalar number One was sometimes called the nonversorand the scaler minus one sometimes called the inversor. These two scalars are special limiting cases corresponding to a versor with an angle approaching that of either zero or pie.

Zero and Pie are then two special scalar points of singularity.

The nonversor and the inversor have the effect when multiplied with vectors of having no effect or of reversing the direction of the vector.

Unlike other versors these two can't be represented by unique arc. The arc of one is a single point. Worse yet minus one can be represented by an infinite number of arcs, because there are an infinite number of shortest lines between two points on the opposite polls of a sphere.

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. 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.

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 have a zero scalar component since the scalar component of a versor is the cosine of the angle of the versor.

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.

Right Quaternion

A right quaternion is quaternion with a scalar component that is zero, $S(q)\; =\; 0$. The angle of a right quaternion is 90 degrees.

Right quaternions may be put in what was called the standard trinomial form. For example if Q is a right quaternion it may be written as:

Q = xi + yj + zk

The study of this important subclass of quaternions called right quaternions, is essentially modern vector analysis.

Product of two Right Quaternion

The product of two Right Quaternions is generally a quaternion. Two very useful operations in Hamilton's calculus were taking the scaler of the product of two Right Quaternion, and taking the Vector of the product of two Right Quaternions.

Let alpha and beta be the right quaternions that result from taking the vectors of two quaternions.

α = V(q₁)

&beta = V(q₁)

Their product in general is then a new quaternion. This product is not ambiguous because classical notation has only one product.

q₃ = αβ

Like all quaternions q₃ may now naturally be decomposed into its vector and scalar parts.

q₃ = V(q₃) + S(q₃)

The terms on the right are called scalar of the product, and the vector of the product of two right quaternions

In hermaphroditical three dimensional notation systems featuring more than one product these two characteristics are often given their own symbols. It should be noted that in some of notation systems, operations related to the scalar of the product, differ in sign from that of the classical operation.

Operators

Ordinal operators

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

Cardinal operations

The two Cardinal operations in classical quaternion notation were geometric multiplication and geometric division and could be written x and ÷

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.

Division

Classical quaternion notation had an operation called division. In fact most classical books on quaternions first introduce the quaternion as the ratio of two vectors. This was sometimes called a Geometric Fraction.

If OA and OB represent two vectors drawn from the origin O, to two other points A and B then the geometric fraction was written as

OA:OB

Alternately if the two vectors are represented by α and β the quotient was written as

α÷β or α/β

Hamilton is already 110 pages into Elements of Quaternions before he even introduces the word quaternion. At the end of article 112 Hamilton reaches the important conclusion he has been working up to: "The quotient of two vectors is generally a quaternion".

Lectures on Quaternions also first introduces the concept of a quaternion as the quotient of two vectors, if

q = α/β.

Logically and by way of definition then

q ×β = α.

Notice that the order of the variables is of great importance. If the order of q and β were to be reversed the result would not in general be α. This is because the product in Hamilton's calculus is not commutative. In other words by definition

α/β = (α) x (1/β)

Again the order of the two quantities on the right hand side of the equation is an important part of the classical definition of division.

Hardypresents the definition of division in terms of pneumonic cancellation rules. "Canceling being performed by an upward right hand stroke".

An important way to think of q is as an operator that changes β into an alpha, by first rotating it, what they used to call an act of version and then changing the length of it, which is what used to be call an act of tension.

Other important operations

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.

Taking the tensor and versor of a quaternion

Another important pair of classical quaternion operations were deconstructing a quaternion into a tensor and versor:

q=Tq.Uq

The formula for the tensor of a quaternion is:

$Tq\; =\; sqrt\{w^2\; +\; x^2\; +\; y^2\; +\; z^2\}\; ,$

Another way to obtain the Tensor of a quaternion is from the common norm.

The common norm of a quaternion is is the product of a quaternion and its conjugate. The square root of the common norm of a quaternion is equal its tensor.

$Tq\; =\; sqrt\{qKq\}\; ,$

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

The taking the axis and angle of a quaternion

Taking the angle of a non-scalar quaternion, resulted in a value greater than zero and less than π.

When a non-scalar quaternion is viewed as the quotient of two vectors, then the axis of the quaternion is a unit vector pointing perpendicular to the plane of the two vectors in this original quotient, in a direction specified by the right hand rule.

In symbols u = Ax.q

The reciprocal of a quaternion

if q = α/β

Then its reciprocal is defined as

1/q = $q^-1$ = β/α

The expression:

q x α x 1/q

Has many important applications for example rotations, particularly when q is the special type of quaternion called a versor. A versor has an easy formula for its reciprocal.

1/(Uq) = S.Uq - V.Uq = K.Uq

The dots between operators show the order to take the operations in, and also help to indicate that S and U for example are two different operations rather than a single operation named SU.

The Common Norm

The product of a quaternion with its conjugate was called the common norm.

The operation of taking the common norm of a quaternion is represented with the letter N The common norm is equal to the square of the tensor of a quaternion. Hence the tensor is generally of greater utility.

In symbols:

qKq = Nq = $(Tq)^2$

Cardinal Operations in Detail

Division in Detail

Division of the Unit Vectors i,j,k

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

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

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

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

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

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

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

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

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

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

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

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

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

Division of two parallel Vectors

While in general the quotient of two vectors is a quaternion, If &alpha and β are two parallel vectors then the quotient of these two vectors is a scalar. For example if

α = ai and β = bi then:
α÷β = α/β = ai/bi = a/b

Where a/b is a scalar.

Division of two non-parallel Vectors

The quotient of two vectors is in general the quaternion:

q = α/β = Tα/Tβ(cosφ + εsinφ)

Where α and β are two non-parallel vectors, φ is that angle between them, and e is a unit vector perpendicular to the plane of the vectors α and β, with its direction given by the standard right hand rule.

Multiplication in Detail

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 scalar.

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

Footnotes

References

• W.R. Hamilton (1853), Lectures on Quaternions, Dublin: Hodges and Smith
• W.R. Hamilton (1899), Elements of Quaternions, 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company.
• A.S. Hardy (1887), Elements of Quaternions
• P.G. Tait (1890), An Elementary Treatise on Quaternions, Cambridge: C.J. Clay and Sons
• Herbert Goldstein(1980), Classical Mechanics, 2nd edition, Library of congress catalog number QA805.G6 1980