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William B. Heard

Rigid Body Mechanics

Mathematics, Physics and Applications

WILEY-VCH Verlag GmbH & Co. KGaA

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William B. Heard


William B. Heard


Mathematics, Physics and Applications

WILEY-VCH Verlag GmbH & Co. KGaA

The Author

William B. Heard

Alexandria, VA

USA

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© 2006 WILEY-VCH Verlag GmbH & Co.

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ISBN-13: 978-3-527-40620-3

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This book is dedicated toPeggy

and the memory of myMother and Father

VII

Contents

Preface XI

1 Rotations 11.1 Rotations as Linear Operators 11.1.1 Vector Algebra 1

1.1.2 Rotation Operators on R3 71.1.3 Rotations Specified by Axis and Angle 81.1.4 The Cayley Transform 121.1.5 Reflections 131.1.6 Euler Angles 151.2 Quaternions 171.2.1 Quaternion Algebra 201.2.2 Quaternions as Scalar–Vector Pairs 211.2.3 Quaternions as Matrices 211.2.4 Rotations via Unit Quaternions 231.2.5 Composition of Rotations 251.3 Complex Numbers 291.3.1 Cayley–Klein Parameters 301.3.2 Rotations and the Complex Plane 311.4 Summary 351.5 Exercises 37

2 Kinematics, Energy, and Momentum 392.1 Rigid Body Transformation 402.1.1 A Rigid Body has 6 Degrees of Freedom 402.1.2 Any Rigid Body Transformation is Composed of Translation and

Rotation 402.1.3 The Rotation is Independent of the Reference Point 41

Rigid Body Mechanics: Mathematics, Physics and Applications. William B. Heard

ISBN: 3-527-40620-4Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

VIII Contents

2.1.4 Rigid Body Transformations Form the Group SE(3) 412.1.5 Chasles’ Theorem 422.2 Angular Velocity 442.2.1 Angular Velocity in Euler Angles 472.2.2 Angular Velocity in Quaternions 492.2.3 Angular Velocity in Cayley–Klein Parameters 502.3 The Inertia Tensor 502.4 Angular Momentum 542.5 Kinetic Energy 542.6 Exercises 57

3 Dynamics 593.1 Vectorial Mechanics 613.1.1 Translational and Rotational Motion 613.1.2 Generalized Euler Equations 623.2 Lagrangian Mechanics 683.2.1 Variational Methods 683.2.2 Natural Systems and Connections 723.2.3 Poincaré’s Equations 753.3 Hamiltonian Mechanics 843.3.1 Momenta Conjugate to Euler Angles 863.3.2 Andoyer Variables 883.3.3 Brackets 923.4 Exercises 98

4 Constrained Systems 994.1 Constraints 994.2 Lagrange Multipliers 1024.2.1 Using Projections to Eliminate the Multipliers 1034.2.2 Using Reduction to Eliminate the Multipliers 1044.2.3 Using Connections to Eliminate the Multipliers 1074.3 Applications 1084.3.1 Sphere Rolling on a Plane 1084.3.2 Disc Rolling on a Plane 1114.3.3 Two-Wheeled Robot 1144.3.4 Free Rotation in Terms of Quaternions 1164.4 Alternatives to Lagrange Multipliers 1174.4.1 D’Alembert’s Principle 1184.4.2 Equations of Udwadia and Kalaba 1194.5 The Fiber Bundle Viewpoint 1224.6 Exercises 127

Contents IX

5 Integrable Systems 1295.1 Free Rotation 1295.1.1 Integrals of Motion 1295.1.2 Reduction to Quadrature 1305.1.3 Free Rotation in Terms of Andoyer Variables 1365.1.4 Poinsot Construction and Geometric Phase 1405.2 Lagrange Top 1455.2.1 Integrals of Motion and Reduction to Quadrature 1455.2.2 Motion of the Top’s Axis 1475.3 The Gyrostat 1485.3.1 Bifurcation of the Phase Portrait 1495.3.2 Reduction to Quadrature 1505.4 Kowalevsky Top 1535.5 Liouville Tori and Lax Equations 1545.5.1 Liouville Tori 1555.5.2 Lax Equations 1565.6 Exercises 159

6 Numerical Methods 1616.1 Classical ODE Integrators 1616.2 Symplectic ODE Integrators 1686.3 Lie Group Methods 1706.4 Differential–Algebraic Systems 1796.5 Wobblestone Case Study 1826.6 Exercises 187

7 Applications 1897.1 Precession and Nutation 1897.1.1 Gravitational Attraction 1907.1.2 Precession and Nutation via Lagrangian Mechanics 1927.1.3 A Hamiltonian Formulation 1957.2 Gravity Gradient Stabilization of Satellites 1977.2.1 Kinematics in Terms of Yaw-Pitch-Roll 1987.2.2 Rotation Equations for an Asymmetric Satellite 1987.2.3 Linear Stability Analysis 2007.3 Motion of a Multibody: A Robot Arm 2037.4 Molecular Dynamics 210

X Contents

Appendix

A Spherical Trigonometry 213

B Elliptic Functions 218B.1 Elliptic Functions Via the Simple Pendulum 218B.2 Algebraic Relations Among Elliptic Functions 222B.3 Differential Equations Satisfied by Elliptic Functions 224B.3.1 The Addition Formulas 224

C Lie Groups and Lie Algebras 225C.1 Infinitesimal Generators of Rotations 225C.2 Lie Groups 227C.2.1 Examples 230C.3 Lie Algebras 231C.3.1 Examples 232C.3.2 Adjoint Operators 234C.4 Lie Group–Lie Algebra Relations 237C.4.1 The Exp Map 237C.4.2 The Derivative of exp A(t) 238

D Notation 241

References 243

Index 247

XI

Preface

This is a textbook on rigid body mechanics written for graduate and advancedundergraduate students of science and engineering. The primary reason forwriting the book was to give an account of the subject which was firmlygrounded in both the classical and geometrical foundations of the subject.The book is intended to be accessible to a student who is well prepared inlinear algebra and advanced calculus, who has had an introductory coursein mechanics and who has a certain degree of mathematical maturity. Anymathematics needed beyond this is included in the text.

Chapter 1 deals with the rotations, the basic operation in rigid body theory.Rotations are presented in several parameterizations including axis angle, Eu-ler angle, quaternion, and Cayley–Klein parameters. The rotations form a Liegroup which underlies all of rigid body mechanics.

Chapter 2 studies rigid body motions, angular velocity, and the physicalconcepts of angular momentum and kinetic energy. The fundamental idea ofangular velocity is straight from the Lie algebra theory. These concepts areillustrated with several examples from physics and engineering.

Chapter 3 studies rigid body dynamics in vector, Lagrangian, and Hamil-tonian formulations. This chapter introduces many geometric concepts asdynamics occurs on differential manifolds and for rigid body mechanics themanifold is often a Lie group. The idea of the adjoint action is seen to be basicto the rigid body equations of motion. This chapter contains many examplesfrom physics and engineering.

Chapter 4 considers the dynamics of constrained systems. Here Lagrangemultipliers are introduced and several ways of determining or eliminatingthem are considered. This topic is rich in geometrical interactions and thereare several examples, some standard and some not.

Chapter 5 considers the integrable problems of free rotation, Lagrange’s top,and the gyrostat. The Kowalevsky top and Lax equations are also considered.Geometrical topics include the Poinsot construction, the geometric phase, and

Rigid Body Mechanics: Mathematics, Physics and Applications. William B. HeardCopyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-40620-4

XII Preface

Liouville tori. Verification and validation are of the utmost importance in theworld of scientific and engineering computing and analytical solutions aretreasured. In addition to the validation service they also play an importantrole in developing our intuitive understanding of the subject, not to mentiontheir intrinsic and historical worth.

The importance of numerical methods in today’s applications of mechan-ics cannot be overstated. Chapter 6 discusses classical numerical methodsand more recent methods tailored specifically for Lie groups. This chapterincludes a case study of a complicated rigid body motion, the wobblestone,which is naturally studied with numerical methods.

The final chapter, Chapter 7, applies the previous material to phenomenaranging in scale from the astronomical to the molecular. The largest scale con-cerns precession and nutation of the Earth. The Earth is nonspherical – anoblate spheroid to first approximation – and the axis of the Earth’s rotation isobserved to move on the celestial sphere. Most of this motion is attributed tothe torque exerted on the nonspherical Earth by the Moon and the Sun andrigid body dynamics explains the effect. Next we study satellite gravity gra-dient stabilization. If Earth’s satellites are not stabilized by some mechanism,they will tumble – as do the asteroids – and will not be useful platforms. Onestabilization mechanism uses the gradient in the Earth’s gravitational fieldand the basics of this mechanism are explained by rigid body theory. On thesame scale, but down to the Earth, we consider the motion of a multibody, amechanical system consisting of interconnected rigid bodies. Rigid body dy-namics is used to study the motion of a robot arm. At the smallest scale weexamine the techniques of molecular dynamics. Physicists and chemists studyproperties of matter by simulating the motions of a large number of interact-ing molecules. At the most basic level this is a problem in quantum mechan-ics. However, there are properties which can be calculated by idealizing thematerial to be a collection of interacting rigid bodies.

Three appendices are provided on spherical trigonometry, elliptic functionsand Lie groups and Lie algebras. Lie groups and Lie algebras unify the subject.Appendix C provides background on all the Lie group concepts used in thetext.

Some choices made in the course of writing the book should be mentioned.The application of mechanics often boils down to making a calculation andgetting the useful number. I have tried to keep calculations foremost inmind. There are no theorem–proof structures in the book. Rather observationsare made and substantiated by methods which are decidedly computational.Rarely are equations put in dimensionless form because one can rely on di-mensional checks to avoid algebra mistakes and misconceived physics. Ofcourse, when it comes time to compute one is well advised to pay attention to

Preface XIII

the scalings. Various notations are used in the text ranging from “old tensor”to intrinsic operators independent of coordinates and the concise notationsof [1,23] which facilitate the computations. There is a glossary in Appendix Dof the notations used. Examples are set aside in italic type and end with thesymbol ♦.

William B. Heard

Alexandria, Virginia, U.S.A., 23 August 2005

1

1Rotations

This chapter is devoted to rotations in three-dimensional space. Rotations arefundamental to rigid body dynamics because there is a one-one correspon-dence between orientations of a rigid body and rotations in three-dimensionalspace. The theory of rotations is a classical subject with a rich history and avariety of modes of expression. We shall begin by expressing the elements ofthe theory in terms of vectors and linear operators. Next, quaternions will beintroduced and additional elements of the theory developed with them. Thiswill lead to some elegant connections between rotations and spherical geom-etry. This leads, via the stereographic projection and Möbius transformations,to a description of rotations in terms of complex variables.

1.1Rotations as Linear Operators

One way to approach rotations is to study their effect on spatial objects. Thelanguage of vectors and matrices provides a natural calculus. This section re-views some basic algebra of vector spaces and establishes our notation. Thenthe angle of rotation and axis of rotation as well as the Euler angles are studiedas ways to parameterize rotation matrices.

1.1.1Vector Algebra

Let us first establish some notation. Let V be a finite-dimensional vector spaceover the real numbers, R. The elements of V, the vectors, will be denoted bybold, lower case Latin letters, u. A basis of V is a set of vectors ei having theproperty that every vector has a unique representation as a linear combinationof basis elements. The basis vectors will be indexed with subscripts. Let V∗ bethe dual vector space of V – the space of all linear, real valued functions on V.The elements of V∗, the covectors, will be denoted by bold, lower case Greek

Rigid Body Mechanics: Mathematics, Physics and Applications. William B. HeardCopyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-40620-4

2 1 Rotations

letters, υ. For each basis ei of V there is a basis εi of V∗ defined by

εi(ej) = δij

where δij is the Kronecker delta. If u = uiei, 1 then e and u will denote

e = ( e1 . . . en ) u =

u1

...un

and the representation of a vector u in the basis e is denoted by

u = eu

The basis e will also be referred to as a frame. Similarly, if υ = υiεi, then ε and

υ will denote

ε =

ε1

...εn

υ =

(υ1 · · · υn

)

and the representation of covector υ in the basis ε is denoted by

υ = υε

This notation is also found in [1] which is a good source of material on geo-metrical aspects of mechanics.

The distinction between a vector u = eu and its components u relative to abasis e is of prime importance. The former is invariant under a change of basisand the latter, of course, is not. We shall always reserve the bold-face font forthe invariant form and the regular typeface components relative to a basis. Alinear operator A on V takes vector x ∈ V to vector A(x) ∈ V and preservesthe operations of the vector space

A(ax + by) = aA(x) + bA(y), for all a, b ∈ R and x, y ∈ V

Given a linear operator A and a basis ei define the matrix representationA = [Ai

j] (row superscript i and column subscript j) by

A(ej) = Aijei

The action of A on any u = ujej is then represented as

Au = Aiju

jei

1) The summation convention that repeated indices indicate a sumover their range is used here and throughout the text.

1.1 Rotations as Linear Operators 3

The matrix can be regarded as operating on a row of basis vectors from theright according to

A(u) = uj f j with f j = Aijei

This can be expressed in a matrix form as

(f1 f2 f3

)=(

e1 e2 e3)

A11 A1

2 A12

A21 A2

2 A23

A31 A3

2 A33

or in the shorthand notationf = eA

Alternatively it can be regarded as operating on a column of components fromthe left according to

A(u) = viei with vi = Aijuj

This can be expressed as

v1

v2

v3

=

A11 A1

2 A12

A21 A2

2 A23

A31 A3

2 A33

u1

u2

u3

orv = Au

Given a pair of vectors u, v, a scalar or inner product on V assigns a non-negative, real number 〈u, v〉 which has the following properties:

if u = 0 then 〈u, u〉 > 0

〈u, v〉 = 〈v, u〉〈aiei, bjej〉 = aibj〈ei, ej〉

An inner product may be expressed in the following equivalent ways:

〈u, v〉 = u · v = (uiei) · (vjej) = uiviGij

where the real numbersGij = ei · ej

are components of a symmetric, positive definite matrix G called the metrictensor. The Euclidean inner product is distinguished by Gij = δij. In the Eu-clidean case

u · v = utv = vtu

4 1 Rotations

Given an inner product we can define the length or norm of a vector andthe angle between two vectors. The norm of u is

‖u‖ =√〈u, u〉

The angle between u and v is

arccos( 〈u, v〉‖u‖‖v‖

)

Tensors are important objects in rigid body mechanics and we now setdown the basics of tensor algebra. We start with the algebraic definition oftensors of rank 2.2 A tensor of rank 2 assigns a real number to a pair of vectorsor covectors and is linear in each argument. A covariant tensor T of rank 2assigns a real number to pairs of vectors T(u, v) ∈ R. The metric tensor is anexample of a covariant tensor. A contravariant tensor T of rank 2 assigns a realnumber to pairs of covectors T(υ, ν) ∈ R. A mixed tensor T of rank 2 assignsa real number to a vector–covector pair T(u, ν) ∈ R. The components of a ten-sor are its values on basis vectors. Thus, a covariant tensor A has componentsaij = A(ei, ej) and a mixed tensor B has components bi

j = B(ei, εj).Tensors can be formed from tensor products of vectors and covectors. The

tensor products are denoted with the symbol⊗ and are defined by their actionon their arguments. Thus we define a covariant tensor υ⊗ ν by υ⊗ ν(u, v) =υ(u)ν(v) and define the mixed tensor u⊗ ν by u⊗ ν(υ, v) = υ(u)ν(v). It isnot the case that every tensor is a tensor product but every tensor is a linearcombination of tensor products of basis vectors. For example,

T(µiεi, vjej) = µiv

jT(εi, ej)

Addition and scalar multiplication of the tensors can be defined by theiraction on vectors

(aT + bS)(u, v) = aT(u, v) + bS(u, v)

In the case of mixed tensors, the result of this construction is a new vectorspace V ⊗V∗. When V has dimension n, V ⊗V∗ has dimension n2 . The sameconstruction can be carried through for pairs of covectors and the resultingvector space V∗ ⊗ V∗ again has dimension n2.

The subspace∧2 V∗ of V∗ ⊗ V∗ consists of skew-symmetric covariant ten-

sors, that is, of covariant tensors Ω which satisfy

Ω(u, v) = −Ω(v, u)

2) The development extends to any rank but we will need only rank 2.


Example 1.1 The tensor T = ν⊗ υ− υ⊗ ν is skew-symmetric because

T(u, v) = ν(u)υ(v)− υ(v)ν(u)

andT(v, u) = ν(v)υ(u)− υ(u)ν(v) = −T(u, v). ♦

The members of∧2 V∗ are called 2-forms over V . Covectors, members

of V∗, are also called 1-forms. There is a product, the wedge product, whichproduces a 2-form ω ∧ µ from two 1-forms ω and µ. The wedge product isdefined by its action on its arguments

ω ∧ µ(u, v) = ω(u)µ(v)−ω(v)µ(u)

The wedge product is basic to the geometric treatment of Hamiltonian me-chanics.

Now we follow [1] to establish the connection between rank 2 mixed tensorsand linear operators. If A is a linear operator, let TA be the tensor defined bythe action TA(v, ν) = ν(A(v)). The components of TA are given by

TAij = TA(ej, εi) = εi(A(ej)) = Ai

j

Thus the components of TA are the same as those of A. We will exploit this cor-respondence, borrowing from the theory of dyadics, to represent linear trans-formations by

A(ukek) = ei Aijε

j(ukek) = ei Aiju

kδjk = ei Aiju

j (1.1)

Thus the action of the second-rank tensor depends on its object. If appliedto a vector–covector pair it produces a scalar and if applied to a vector it re-turns another vector. This is reminiscent of the dot and double-dot productsof dyadics [2] which are closely related to second-rank tensors.

The combinations εe and e⊗ ε arise frequently in the manipulation of ten-sors. The product εe is the identity matrix because

ε1

ε2

ε

([e1e2e3

])= [εi(ej)] = I

The product e⊗ ε is the identity operator because

[e1e2e3

]⊗

ε1

ε2

ε

= ei ⊗ εi

and for any vector v = viei

(ei ⊗ εi)(vkek) = vkeiδik = v

6 1 Rotations

The above representation leads to a succinct representation of linear opera-tors

A = eAε

which is shorthand for A = Aijei ⊗ εi so that

A(v) = eAε(ev) = eAv

Now consider the effect of a change of basis. Suppose e and e are bases of Vlinearly related by

e = eB

Then a vector v has the representations

v = ev = eBv = ev

so that the components in the two bases are related by

v = Bv

The covector relationsI = eε = eBε = eε

giveε = Bε

Thus a covector µ has the representations

µ = µε = µB−1ε = µε

so that the components in the two bases are related by

µ = µB

The effect of change of basis on a linear operator follows immediately

A = eAε = eBAB−1ε = eAε

orA = BAB−1

Given the linear operators A, B with the matrix representations A, B, the ma-trix representation of the composite map B A,A followed by B, is the matrixproduct BA. The basis vectors transform as

e′ = eBA

and the components transform as

v′ = BAv


For any n the vector space Rn has the standard basis ei where ei is thecolumn of length n having a single nonzero entry, 1 in row i. The inner productin the standard basis is

u · v = ut = vu = uivi

where no distinction is made between ui and ui. In R3 there is also a vectorproduct or cross product

(aiei)× (bjej) = εijkaibjek = (a2b3− a3b2)e1 +(a3b1− a1b3)e2 +(a1b2− a2b1)e3

where εkij is the permutation symbol

εkij =

1 if i j k is an even permutation of 123−1 if i j k is an odd permutation of 123

0 otherwise

We will have no further need in this chapter to distinguish between subscriptsand superscripts, so subscripts will be used. In this case matrix entries will bedenoted by Aij with row index i and column index j. Superscripts will returnwith a vengeance when generalized coordinates are considered.

1.1.2Rotation Operators on R3

A rotation is a linear transformation, R, that fixes the origin, preserves thelengths of vectors, and preserves the orientation of bases. That is,

R : V → V : x → R(x)

R(0) = 0

R(x) · R(x) = x · x

R(e1) · (R(e2)×R(e3)) = e1 · (e2 × e3)

The length-preserving property becomes, in a matrix form,

R(x) · R(x) = (xjRijei) · (xl Rklek) = xjxl RijRil = xixi for all xi ∈ R3

which implies thatRijRil = δjl or RtR = I

This defines an orthogonal matrix. The orientation preserving condition be-comes

Rk1εkijRi2Rj3 = 1 or det R = +1

When R and S are orthogonal, then (RS)tRS = StRtRS = I. When R is or-thogonal R−1 = Rt and (R−1)tR−1 = RRt = I. Therefore RS and R−1 are

8 1 Rotations

also orthogonal. Clearly I is orthogonal. The product of orthogonal matri-ces preserves the determinant, that is, if det R = det S = 1 then det RS =det R det S = 1. If det R = 1 then det R−1 = 1 because det RR−1 = det I = 1.

It follows from these facts that the orthogonal matrices of determinant 1form a group.3 The group is called SO(3) – the special orthogonal group of or-der 3. This group has the additional structure of a three-dimensional manifoldand is therefore a Lie group. In the next section we begin to study parame-terizations, or coordinates, of SO(3). The basics of the Lie group theory areoutlined in Appendix C.

In addition to preserving lengths, orthogonal matrices preserve angles be-cause they preserve inner products

(Ru)tRv = utRtRv = utv

Members of SO(3) also preserve R3 vector products in the sense thatA(x× y) = Ax × Ay. To prove this, it is enough to show it for e1, e2, e3, thestandard basis of R3. Let A ∈ SO(3) be presented in terms of its orthonormalcolumns, A = [c1c2c3] so that Aei = ci. Then

A(ei × ej) = A(εijkek) = εijkAek = εijkck = ci × cj = Aei × Aej

Every member of SO(3) fixes not only the origin but actually an entire line.This follows from the structure of eigensystems of rotation matrices. Thelength preserving property of a rotation requires that eigenvalues have mag-nitude 1. To show this we must allow for complex eigenvalues and eigenvec-tors and use the norm ‖x‖2 = (x∗)tx where x∗ is the complex conjugate of x.Then Rx = λx implies that ‖Rx‖2 = λ∗λ(x∗)tx = |λ|2‖x‖2. In other words‖Rx‖ = ‖x‖ implies |λ| = ±1. The characteristic polynomial of a rotationmatrix is a cubic and one of its roots must be +1 because det R = 1. Thus,any eigenvector corresponding to λ = 1 is fixed and real. The set of all sucheigenvectors forms the axis of rotation.

1.1.3

Rotations Specified by Axis and Angle

First consider plane rotations. Represent vectors as complex numbers (x, y) ↔x + ıy = ρ exp(ıθ). Then a counterclockwise rotation by angle φ is simplymultiplication by exp(ıφ): z → exp(ıφ)z = ρ exp[ı(θ + φ)]. In rectangularcomponents

x + ıy = z → z′ = eıθ(x + ıy) = (x cos θ− y sin θ) + ı(x sin θ + y cos θ)

3) A group G is a set equipped with a binary operation such that ifa, b ∈ G then ab ∈ G. There is an identity element e such that ea = afor every a ∈ G and for every a ∈ G there is an inverse a−1 such thataa−1 = e.


This shows that rotations by angle θ about the x, y, z axes are represented by

1 0 00 c −s0 s c

c 0 s0 1 0−s 0 c

and

c −s 0s c 00 0 1

respectively, where c = cos θ, s = sin θ.These matrices have alternative expressions as operators which emphasize

the role of the axis of rotation and the rotation angle. To state the alternativeforms we first introduce the idea of duality between R3 and so(3), which is theset of all skew-symmetric 3× 3 matrices, that is matrices which have the prop-erty that At = −A. In fact so(3) is the Lie algebra of the Lie group SO(3) (seeAppendix C). To each vector v there corresponds a skew-symmetric matrix v,

v =

v1v2v3

←→

0 −v3 v2v3 0 −v1−v2 v1 0

= v

and to each skew-symmetric matrix M there is a vector M,

M =

0 −u3 u2u3 0 −u1−u2 u1 0

←→

u1u2u3

= M

In the language of Lie algebra v = adv (see Section C.3.2).Using the canonical basis e1, e2, e3 = i, j, k, this is expressed in terms

of tensor products as

u1u2u3

←→ u1 i + u2 j + u3k

where

i = k⊗ j− j⊗ k j = i⊗ k− k⊗ i k = j⊗ i− i⊗ j

are the invariant forms of the operator. Here we have identified R3 and(R3)∗ via

u1u2u3

←→

(u1 u2 u3

)

oru⊗ v ←→ uvt

10 1 Rotations

With these preliminaries we can write that the rotations about the coordinateaxes are

i⊗ i + c(I − i⊗ i) + si

j⊗ j + c(I − j⊗ j) + sj

k⊗ k + c(I − k⊗ k) + sk

These expressions can be generalized to an arbitrary axis of rotation deter-mined by the unit vector n. The form of the expressions suggests that onemight construct a basis containing n and write immediately that

Rn(θ) = n⊗ n + cos θ(I − n⊗ n) + sin θn

and we now proceed to justify this. Let P represent the operator relative to thestandard basis which projects a vector onto the axis of rotation (Fig. 1.1)

P = nnt

P has the property that, for any r, Pr is a multiple of n

Pr = nntr = λn λ = ntr

and P is idempotent

P2 = (nnt)(nnt) = n(ntn)nt = nnt = P

I − P then represents the operator which projects a vector onto the plane nor-mal to n because for any r

nt(I − P)r = ntr− ntnntr = 0

and(I − P)2 = I − P− P + P2 = I − P

Let N = n. Any rotation of r through an angle θ leaves Pr invariant androtates (I − P)r by an angle θ in the plane spanned by (I − P)r, n× r. Thusthe rotation is given by r′ = Rn(θ)r with

Rn(θ) = nnt + cos θ(

I − nnt)+ sin θ N

It is easy to show that the operators P, I− P, N form a closed system definedby Table 1.1. Here we sketch the proof of one entry as an illustration

N2r = n× (n× r) = (r · n)n− r = (P− I)r

The relations in Table 1.1 can be used to recast the rotation operator entirelyin terms of N,

R = I + sin θN + (1− cos θ)N2 (1.2)


Fig. 1.1 Rotation about the axis n by an angle θ.

Table 1.1 Composition of projection operators P = nnt, I − P, and N = n.

P I − P NP P 0 0

I − P 0 I − P NN 0 N −(I − P)

P N N2

P P 0 0N 0 N2 −NN2 0 −N −N2

We close this section with a derivation of the general rotation matrix usinggeneral principles of linear algebra. A general maxim in linear algebra is thatproblems should be worked in the “right” basis, i.e., the basis in which theproblem assumes its simplest, most revealing form. Since rotations are givenby an axis, n, and an angle, α, the “right” basis in which to study rotationswould naturally include n. Accordingly, let b1 = n; choose unit vector b2⊥nand b3 = b1 × b2. In this basis we can write immediately

Rn(α) =

1 0 00 c −s0 s c

where c = cos α and s = sin α and α is reckoned from b2 in the b2–b3 plane.To work in another basis – the standard basis (e1, e2, e3), for example – we canuse the change of basis relations

R = bRn(α)β = eARn(α)Atε = eR′ε

A is the change of basis matrix such that

b = eA Aij = bj · ei

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