|
|
|
|
Classical Hamiltonian quaternions
|
| |
|
| |
This article is about quaternions, a mathematical entity defined by William Rowan Hamilton. Its primary source is Elements of Quaternions a book first published shortly after Hamilton's death based entirely on his notes, proof sheets, and a typed manuscript. The other primary source is Lectures on Quaternions, a book written by Hamilton based on his lectures on the subject shortly after his discovery of the quaternion.
Discussion
Ask a question about 'Classical Hamiltonian quaternions' Start a new discussion about 'Classical Hamiltonian quaternions' Answer questions from other users |
Encyclopedia
This article is about quaternions, a mathematical entity defined by William Rowan Hamilton. Its primary source is Elements of Quaternions a book first published shortly after Hamilton's death based entirely on his notes, proof sheets, and a typed manuscript. The other primary source is Lectures on Quaternions, a book written by Hamilton based on his lectures on the subject shortly after his discovery of the quaternion. The many outstanding books on the subject of quaternions, all based very closely on the thinking of Hamilton and written before 1901 are the only proper secondary sources for this article. This article excludes secondary sources written after 1901. The choice of the year 1901 is a somewhat artificial agreed upon editorial device need because the subject of quaternions now spans three centuries. The quaternion defined by Hamilton is a modern quaternion, however over the centuries some writers on the subject have broken with Hamilton's original definitions, some drastically. For a discussion of quaternions, that includes not only the quaternions defined by Hamilton, but other definitions as well a good place to start is the main article on quaternions. The story of the many colorful people involved in the development of quaternions before 1901 is an interesting one and can be found in the article history of quaternions, however this article is about a Mathematical entity, not about the people who developed it.
Many modern readers of Nineteenth-century works on the calculus of quaternions find difficulties with some works on the subject written after 1901 because the notation used by classical writers, based on the standard notation and vocabulary of Hamilton is different from that of some modern writers. Modern writers also differ from each other greatly in their choice of notation and definitions. Some of the most popular, and reasonably priced modern paper back books on quaternions, released after 2005 are new editions of hard cover books written more than a century ago. Old sources have the added appeal that they can be viewed in their entirety online. Almost all primary and secondary citations in this article have hot links into the original classical texts that can be viewed online.
Some modern books use the word quaternion but use definitions, notation and terminology so different from Hamilton's that according to the precise definition of a quaternion offered in this article, these modern books are not about quaternions. If they are about some sort of quaternion, certainly they are not the classical Hamiltonian quaternions.
Modern readers who have never read a book on quaternions, are often confused on first reading a 19th century classic on the subject, because they have misconceptions about the subject due to having read books that use the word quaternion but are about something else. Once they understand the correct definitions of these jointly used terms, this difficulty soon fades.
Short history
In classical quaternion notation a unit of distance squared was equal to a negative scalar quantity. The Pythagorean theorem, where B = 3i and C = 4j are the sides of a right triangle and A is the hypotenuse would look like:
-
-
To put it in classical quaternion terminology: the square of every vector is a negative scalar.1n 1835 years before Hamilton discovered quaternions he wrote an article entitled Algebra the Science of Pure Time, in which he expressed the view that time worked like a real number or scalar.
Hamilton's discovery of quaternions has sometimes been linked a 19th century version of the modern concept of spacetime. In the words of John Baez, Hamilton gave quaternions a "cosmic significance". As time progressed, Hamilton devoted more and more of his efforts to pure mathematics, continuing his research into quaternions from the time of his discovery of them until his death in 1865.
Classical elements of a quaternion
Tensor
At a basic level it is alright to think a tensor as just a positive number. Mathematically it functions exactly like a positive number. If you add two tensors you get another tensor, just like when you add two positive numbers. The same is generally true with multiplication and division. On a more technical level Hamilton's classical definition of a tensor is of a signless number. In other words a number with out a plus or minus sign in front of it. 5 is an example of a tensor and so is 2. A plus sign is an operator, and you can put an operator inbetween two tensors and perform the operation of addition producing a new tensor. Zero is the problem child of the tensors, if you divide by zero the answer is infinite. On the other hand if you add zero to another tenser you get the same number you had before. For example 0 + 2 = 2. According to Hamilton's notation it is OK to just leave off the zero, write in shorthand +2 = 2. When two tensors are subtracted the answer may not be a positive number, and hence not a tensor, but a new and different kind of number called a scalar. For example 2 - 5 = -3. Minus two is not a tensor. Actually -3 like any negative number can be written as 0 - 3. Hamilton asks you to agree that there is really no need to write a zero in front of the three, with the understanding that -3 is really just short hand for 0 - 3. In the classical treatment of the subject it is skipping ahead to say that
the tensor of a quaternion is its magnitude. Think of it as the "stretching factor", the amount by which the application of the quaternion lengthens a quantity. It can be proven that tensor is the square root of the norm — this is a one-dimensional quantity, quite distinct from the modern sense of tensor, coined by Woldemar Voigt in 1898 to express the work of Riemann and Ricci.
The product of two tensors is another tensor, and the quotient of two non-zero tensors is another tensor.
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
Every quaternion can be decomposed into a scalar and a vector.
These two operations S and V were called take the Scalar of and take the vector of a quaternion. The vector part of a quaternion is also called the right part..
In abridged notation parenthesis are not required and were not normally used. In the above expression Vq and Sq could be written with out ambiguity. The operation of taking the vector of a quaternion took priority over the operation of raising to a power, unless a dot was placed between the operation and the rest of the expression, as in the relations below.
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.
Hamilton introduced the world to the concept of a vector in the 1840s. In Hamilton's first lecture article 15, he introduces the word vector, from the Latin vection, or to move.
Versor
The versor of a quaternion which can be written as
is another 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.
When a versor and a vector which lies in the plane of the versor are multiplied the result is a new vector of the same length but turned by the angle of the versor.
Right versor
When the arc of a versor has the magnitude of a right angle, then it is called a right versor,a right radial 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
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. If the two vectors in the quotient are at right angles to each other then the quaternion is called a right radial quotient.
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 pi.
Zero and Pi 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, . 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:
The study of this important subclass of quaternions called right quaternions, is essentially modern vector analysis. It can be proven that every vector function is a function of a right quaternion.
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.
Their product in general is then a new quaternion represented here by r. This product is not ambiguous because classical notation has only one product.
Like all quaternions r may now naturally be decomposed into its vector and scalar parts.
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
Alternately if the two vectors are represented by a and ß the quotient was written as
or
Hamilton spends a great deal of time on the development of the concept of a vector and 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
Logically and by way of definition then
.
Notice that the order of the variables is of great importance. If the order of q and ß were to be reversed the result would not in general be a. This is because the product in Hamilton's calculus is not commutative. The quaternion q can be thought of as an operator that changes ß into a, 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.
Also by definition the quotient of two vectors is equal to the numerator times the reciprocal of the denominator. Since multiplication of vectors is not comutative the order can not be changed in the following expression.
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".
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. Here S and V are operators acting on q. Parenthesis can be omitted in these kinds of expressions with out ambiguity.
In the classical era this is what the notation looked like:
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:
The formula for the tensor of a quaternion is:
Another way to obtain the tensor of a quaternion is from the common norm, defined as the product of a quaternion and its conjugate. The square root of the common norm of a quaternion is equal to its tensor.
Taking the conjugate
The K(q) operator means, take the conjugate. The conjugate of a quaternion is another quaternion obtained by multiplying the vector part of the first quaternion by minus one.
If
then
.
The expression
,
means, assign the quaternion r the value of the conjugate of the quaternion q.
Axis and angle of a quaternion
Taking the angle of a non-scalar quaternion, resulted in a value greater than zero and less than p.
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. The angle is the angle between the two vectors.
In symbols
Reciprocal of a quaternion
if
Then its reciprocal is defined as
The expression:
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.
In words this says that the reciprocal of a versor is equal to its conjugate. 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.
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. By definition the common norm is the product of a quaternion with its conjugate. It can be proven that common norm is equal to the square of the tensor of a quaternion. Hence the tensor is generally of greater utility.
In symbols:
The common norm of a versor is always equal to positive unity.
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.
Division of two parallel Vectors
While in general the quotient of two vectors is a quaternion, If a and ß are two parallel vectors then the quotient of these two vectors is a scalar. For example if
,
and then
Where a/b is a scalar.
Division of two non-parallel Vectors
The quotient of two vectors is in general the quaternion:
Where a and ß are two non-parallel vectors, f is that angle between them, and e is a unit vector perpendicular to the plane of the vectors a and ß, 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.
Using the quaternion multiplication table we have:
Then collecting terms:
The first three terms are a scalar.
Letting
So that the product of two vectors is a quaternion, and can be written in the form:
See also
|
| |
|
|
