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Physical Motions and Quaternions

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The representation of physical motions by various types of quaternions D. H. Delphenich † Abstract: It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex, and complex dual quaternions, respectively. It is shown how someof the physically-useful tensors and spinors can be represented by the various kinds of quaternions. The basic notions of kinematical states are described in each case, except complex dual quaternions, where their possible role in describing the symmetries of the Maxwell equations is discussed. † E-mail: [email protected] , Website: neo-classical-physics.info. CONTENTS Page INTRODUCTION……………………………………………………............. 1 References………………………………………………………………… 6 I. ALGEBRAS 1. General notions……………………………………………………………. 10 2. Ideals in algebras…………………………………………………………... 15 3. Automorphisms of algebras………………………………………..……… 17 4. Representations of algebras……………………………………………….. 18 5. Tensor products of algebras……………………………………………….. 21 References………………………………………………………………… 23 II. REAL QUATERNIONS 1. The group of Euclidian rotations………………………………………… 24 2. The algebra of real quaternions………………………………………….. 30 3. The action of rotations on quaternions…………………………………… 35 4. The kinematics of fixed-point rigid bodies………………………………. 41 References……………………………………………………………….. 47 III. DUAL QUATERNIONS 1. The group of Euclidian rigid motions…………………………………… 48 2. The algebra of dual numbers……………………………………………. 54 3. Functions of dual numbers………………………………………………. 56 4. Dual linear algebra………………………………………………………. 56 5. The algebra of dual quaternions…………………………………………. 64 6. The action of rigid motions on dual quaternions………………………… 69 7. Some line geometry……………………………………………………… 71 8. The kinematics of translating rigid bodies………………………………. 72 References……………………………………………………..………… 75 IV. COMPLEX QUATERNIONS 1. Functions of complex variables………………………………………….. 76 2. The group of complex rotations…………………………………………. 77 3. The algebra of complex quaternions…………………………………….. 81 4. The action of the Lorentz group on complex quaternions………………. 88 5. Some complex line geometry……………………………………………. 98 6. The kinematics of Lorentzian frames……………………………………. 101 References…………………………………………………..…………… 105 Table of Contents ii V. COMPLEX DUAL QUATERNIONS 1. The group of complex rigid motions…………………………………… 107 2. The algebra of complex dual numbers…………………………………. 109 3. Functions of complex dual numbers……………………………………. 111 4. Complex dual linear algebra……………………………………………. 112 5. The algebra of the complex dual quaternions………………………….. 115 6. The action of the group of complex rigid motions on complex dual quaternions……………………………………………………………... 121 7. The role of complex rigid frames in physics…………………………… 122 INTRODUCTION According to Wilhelm Blaschke [ 1], the first vestiges of the concept of a quaternion went to back to the work of Leonhard Euler on rigid-body rotations in 1776 [ 2]. Olinde Rodrigues seems to have developed much of the basics of the subject in 1840, without actually introducing the term “quaternion.” However, the first major attempt to develop the concept in its own right, along with its applications to physics, seems to have the posthumously-published book of Sir William Rowan Hamilton that he called Elements of Quaternions [ 3] that came almost a century after Euler in 1866, although the work was done starting in 1843. Hamilton envisioned quaternions as a geometric algebra that would extend the three- dimensional algebra of the vector cross product to a four-dimensional algebra in which the Euclidian scalar product also played a role in defining the product of its elements. Thus, the algebra would be closely related to rotations by the fact that the vector cross product defines a Lie algebra on R3 that is isomorphic to so (3; R), which consists of infinitesimal generators of Euclidian rotations, as well as the fact that rotations preserve the Euclidian scalar product. In that monumental treatise, the coefficients of quaternions were treated as if they could be either real or complex, as it suited the purpose, although Hamilton suggested that the complex kind might be referred to as “biquaternions.” However, since the work predated the theory of relativity, no mention was made of the role that the complex quaternions might play in regard to relativistic motions. Apparently, the subsequent history of the theory of quaternions was somewhat marginal to the mainstream of both mathematics and physics for quite some time. However, from 1899 to 1913 there was an international Quaternion Society that formed around Alexander MacFarlane that was dedicated to keeping the concept alive. For the most part, their main acceptance in the early years seems to have been with the algebraists (cf., e.g., Shaw [ 4]), who saw them as a useful example of an algebra. However, later researchers in physics and mechanical engineering continued to expand upon the usefulness of quaternions in applications that involved the representation of three-dimensional Euclidian rotations. For instance, to this day, the inertial navigation community regards the representation of kinematics by real quaternions as being their most computationally efficient algorithm for the propagation of rotating frames. Although the method of Euler angles requires one less coordinate to describe a rotation, nevertheless, the differential equations that one must integrate involve products of trigonometric functions, which tends to slow down the computational speed considerably. Although the method of direction cosine matrices only involves basic arithmetic operations on the components of the matrices, nonetheless, they are matrices with nine components, which also tends to slow things down. Thus, the four components of a unit quaternion seem to provide an ideal compromise. In a different part of geometry and mechanics, geometers such as Louis Poinsot [ 5] and Michel Chasles [ 6] were exploring the geometry of the larger group of rigid motions, which also includes translations, in addition to the rotations; both two and three dimensional rigid motions were dealt with in detail. The geometry that they were Introduction 2 considering was actually projective geometry, not affine Euclidian geometry. Chasles found that the concept of an infinitesimal center of rotation for a planar rigid motion could be extended to a “central axis” for any three-dimensional rigid motion, which allowed one to decompose the motion into a rotation about the axis and a translation along it. Previously, in the context of statics, Poinsot found that a finite set of spatially- distributed force vectors that acted on a rigid body could be replaced with an equivalent force-moment about a central axis and a force along it. The German geometers Julius Plücker [ 7] and his illustrious student Felix Klein [ 8] expanded upon the projective- geometric nature of these constructions, and introduced the term “Dyname” for a finite spatial distribution of force vectors. (The French were using the term “torseur,” which eventually became the more modern term “torsor.”) In the meantime, Sir Robert Ball [ 9] had introduced the term “screw” to describe the canonical form of the rigid motion and “wrench” to describe the canonical form of a force distribution. To some extent, the German school of geometrical kinematics culminated in the 1903 treatise [ 10 ] of another student of Plücker named Eduard Study that was entitled Die Geometrie der Dynamen . One of the innovations that Study introduced in that work was the algebra of “dual numbers,” which he applied to study of quaternions to produce “dual quaternions.” Actually, William Kingdon Clifford had previously sketched out a theory [ 11] of what he was calling “biquaternions,” although his usage of that word was inconsistent with that of Hamilton, since Clifford’s biquaternions were quaternions whose components were dual numbers, while Hamilton’s biquaternions had complex components. However, the treatise of Study developed the concepts in much more detail than that of Clifford. Since dual quaternions represent an algebra over R8, some authors (e.g., MacAulay [ 12 ]) referred to them as “octonions.” Unfortunately, that usage is inconsistent with the modern usage of that term to refer to Cayley numbers, which defines a division algebra over R8, unlike dual quaternions, which have divisors of zero, and therefore cannot be a division algebra. The dual numbers represent an algebra over R2, just as the complex numbers do, but the difference is that the basic object of the dual number algebra is a symbol ε that is nilpotent – viz., ε2 = 0 – while the basic object of the complex algebra is a symbol i with the property that i2 = −1. The effect of introducing ε is to produce an algebra that is not a division algebra – in particular, it has divisors of zero – with the property that the product of two dual numbers a + εb and c + εd is ac + ε(ad + bc ), which then combines the usual multiplication of real numbers with their addition. Although it is not obvious at this stage of the discussion, when one uses such numbers for the components of quaternions, it allows on to represent three-dimensional rigid motions by means of the group of unit dual quaternions, just as the unit quaternions carry a representation of the group of three- dimensional Euclidian rotations; in fact, it is the spin representation that is isomorphic to SU (2). Apparently,
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