Spherical Trigonometry - Springer978-0-306-47122-3/1.pdf · Spherical Trigonometry 663 bivector of...

Appendix A

Spherical Trigonometry

In Section 2-4 we saw how efficiently geometric algebra describes relationsamong directions and angles in a plane. Here we turn to the study of suchrelations in 3-dimensional space. Our aim is to see how the traditional subjectsof solid geometry and spherical trigonometry can best be handled withgeometric algebra. Spherical trigonometry is useful in subjects as diverse ascrystalography and celestial navigation.

First, let us see how to determine the angle that a line makes with a planefrom the directions of the line and the plane. The direction of a given line isrepresented by a unit vector while that of a plane is represented by a unitbivector The angle between and is defined by the product

where i is the unit righthanded trivector, is a unit vector and Thevector and trivector parts of (A.1) are

Our assumption that i is the unit righthanded trivector fixes the sign ofso, by (A.2b), is positive (negative) when is righthanded (left-handed). The angle α is uniquely determined by (A.1) if it is restricted to therange Equations (A.2a, b) are perfectly consistent with the conventional in-

terpretation of and as components of projection and rejection, asshown in Figure A.1. Thus, according to the definition of projection by Equation(4.5b) of Section 2-4, we have

We can interpret this equation as follows. First the unit vector is projectedinto a vector with magnitude Right multiplication by thenrotates the projected vector through a right angle into the vector whichcan be expressed as a unit vector times its magnitude Thus, the

661

662 Appendix A

product is equivalent to a projection of into the plane followed by arotation by

To interpret the exponential in (A.1), multiply (A.1) by to get

This expresses the unit bivector as theproduct of orthogonal unit vectors andThe factor has rotated into the plane.We recognize as a spinor which rotatesvectors in the plane through an angleThe unit vector specifies the axis of rota-tion.

Now let us consider how to represent theangle between two planes algebraically. Thedirections of two given planes are representedby unit bivectors and We define the dihedral angle c between and bythe equation

Both the magnitude of the angle (with ) and the directionof of its plane are determined by (A.3). Separating (A.3) into scalar andbivector parts, we have

Note that for c = 0, Equation (A.3) becomes so since

The geometrical interpretation of (A.3) is indicated inFigure A.2, which shows the two planes intersecting in aline with direction The vector is therefore a com-mon factor of the bivectors and Consequentlythere exist unit vectors and orthogonal to such that

Note that the order of factors corresponds to the orien-tations assigned in Figure A.2. The common factorvanishes when and are multiplied;

Note that this last equality has the same form as Equation (4.9) with the unit

Fig. A.1. The angle between a vec-tor and a bivector.

Fig. A.2. Dihedral Angle.Note that the orientation of

is chosen so it opposesthat of if is brought intocoincidence with by ro-tating it through the anglec.

and have the factorizations

Spherical Trigonometry 663

bivector of the plane expressed as the dual of the unit vector Itshould be clear now that (A.4b) expresses the fact that is a bivectordetermining a plane perpendicular to the line of intersection of the and

planes and so intersecting these planes at right angles.

Fig. A.3. A spherical triangle. Fig. A.4. Altitudes of a sphericaltriangle.

Now we are prepared to analyze the relations among three distinct direc-tions. Three unit vectors can be regarded as vertices of a sphericaltriangle on a unit sphere, as shown in Figure A.3. The sides of the triangle arearcs of great circles determined by the intersection of the circle with theplanes determined by each pair of vectors. The sides of the triangle havelengths A, B, C which are equal to the angles between the vectors Thisrelation is completely described by the equations

where The unit bivectors are directionsfor the planes determining the sides of the spherical triangle by intersectionwith the sphere. Of course, we have the relations

as well as

The angles a, b, c of the spherical triangle in Figure A.3 are dihedral anglesbetween planes, so they are determined by equations of the form (A.3),namely,

664 Appendix A

whereEquations (A.5a, b, c) and (A.7a, b, c) with (A.6a, b) understood deter-

mine all the relations among the sides and angles of a spherical triangle. Letus see how these equations can be used to derive the fundamental equations ofspherical trigonometry. Taking the product of Equations (A.5a, b, c) andnoticing that

we get

This equation can be solved for any one of the angles A, B, C in terms of theother two. To solve for C, multiply (A.8) by to get

If the exponentials are expanded into scalar and bivector parts and (A.7a) isused in the form then (A.9) assumes the expanded form

Separating this into scalar and bivector parts, we get

Equation (A.10) is called the cosine law for sides in spherical trigonometry. Itrelates three sides and an angle of a spherical triangle and determines any oneof these quantities when the other two are known.

Since the value of C can be determined from A and B by (A.10), thedirection is then determined by (A.11). Thus equations (A.10) and (A.11)together determine C from A and B. Nothing resembling Equation (A.11)appears in traditional spherical trigonometry, because it relates directions

whereas the traditional theory is concerned only with scalar relations.Of course, all sorts of scalar relations can be generated from (A.11) bymultiplying by any one of the available bivectors, but they are only ofmarginal interest. The great value of (A.11) is evident in our study ofrotations in 3 dimensions in Section 5-3.

We can analyze consequences of (A.7a, b, c) in the same way we analyzed(A.5a, b, c). Observing that


we obtain from the product of Equations (A.7a, c, b)

This should be compared with (A.8). From (A.12) we get

The scalar part of this equation gives us

This is called the cosine law for angles in spherical trigonometry. Obviously, itdetermines the relation among three angles and a side of the sphericaltriangle.

The cosine law was derived by considering products of vectors in pairs, sowe may expect to find a different “law” by considering the productInserting from (A.5b) and the corresponding relations from(A.5c, a) into we get

We can find an analogous relation from the product Using (A.7b), weascertain that

Obtaining the corresponding relations from (A.7a, b), we get

The ratio of (A.15) to (A.16) gives us

This is called the sine law in spherical trigonometry. Obviously, it relates anytwo sides of a spherical triangle to the two opposing angles.

We get further information about the spherical triangle by considering theproduct of each vector with the bivector of the opposing side. Each producthas the form of Equation (A.1) which corresponds to Figure A.1. Thus, wehave the equation

The angles are “altitudes” of the spherical triangle with lengthsas shown by Figure A.4. If the trivector parts of (A.18a, b, c) are

substituted into (A.15) and (A.16) we get the corresponding equations oftraditional spherical trigonometry:

666 Appendix A

This completes our algebraic analysis of basic relations in spherical trigo-nometry.

Exercises

(A.1) Prove that Equation (A.8) is equivalent to the equations

and

(A.2) The spherical triangle satisfying Equations (A.5) and (A.7)determines another spherical triangle by the dualityrelations

The triangle is called the polar triangle of the trianglebecause its sides are arcs of great circles with and as

poles.Prove that the sides of the polar triangle are equal to

the exterior angles supplementary to the interior angles ofthe primary triangle, and, conversely, that the sides A, B, C of theprimary triangle are equal to the exterior angles supplementary tothe interior angles of the polar triangle.

From Equations (A.18a, b, c) prove that corresponding altitudesof the two triangles lie on the same great circle and that the distancealong the great circle between and is

(A.3) For the right spherical triangle with prove that

(A.4) Prove that an equilateral spherical triangle is equiangular with anglea related to side A by

(A.5) Assuming (A.5a, b, c), prove that


Hence,

Use this to derive the sine law, and, from (A. 19), expressions forthe altitudes of a spherical triangle.

(A.6) Prove that

Assuming (A.5a, b, c) use the identity

to prove that

Note that this can be used with (A. 19) to find the altitudes of aspherical triangle from the sides.

(A.7) Find the surface area and volume of a parallelopiped with edges oflengths a, b, c and face angles A, B,

(A.8) Establish the identity

Note that the three terms differ only by a cyclic permutation of thefirst three vectors. Use this identity to prove that the altitudes of aspherical triangle intersect in a point (compare with Exercise2–4.11a).

(A.9) Prove that on a unit sphere the area of a spherical triangle withinterior angles a, b, c (Figure A.3) is given by the formula

Since is given by the differencebetween the sum of interiorangles for a spherical triangleand a plane triangle, it is oftencalled the spherical excess.Hint: The triangle is determinedby the intersection of great cir-cles which divide the sphere intoseveral regions with area

or as shown in Figure A.5.What relation exists between theangle a and the area Fig. A.5. Spherical excess.

Appendix B

Elliptic Functions

Elliptic functions provide general solutions of differential equations with theform

where f(y) is a polynomial in Such equations are very common inphysics, arising frequently from energy integrals where the left side of theequation comes from a kinetic energy term.

Since different polynomials can be related by such devices as factoring andchange of variables, it turns out that the general problem of solving (B.1) fora large class of polynomials can be reduced to solving a differential equationof the standard form

for and The solution of this equation for the condi-tions

is denoted by

(B.4).

(Pronounced “ess-en-ex”). Of course, this function depends on the value ofthe parameter k, which is called the modulus.

Direct integration of (B.2) produces the inverse function

668

Elliptical Functions 669

It is an odd function of y, which increases steadily from 0 to

as y increases from 0 to 1. Consequently, is an odd function of x, andit has period 4K; that is

The integral (B.6) is called a complete elliptic integral of the first kind. We canevaluate it by a change of variables and a series expansion:

The function K(k) is graphed in Figure B.2.Two other functions cn x and dn x can be defined by the equations

along with the condition that their deriva-tives be continuous to determine the signof the square root. Since dn x isalways positive with period 2K, while cn xhas period 4K.

The three functions sn x, cn x and dn xare called Jacobian elliptic functions, orjust elliptic functions. They may be re-garded as generalizations or distortions ofthe familiar trigonometric functions. In-deed, from the above relations it is read-ily verified that for

and for

Traditionally, the nomenclature of elliptic functions is used only when k is inrange 0 < k < 1.

Graphs of the elliptic functions are shown in Figure B.2. Tables of ellipticfunctions can be found in standard references such as Jahnke and Emde

Fig. B.1. Graph of the period 4K as a func-tion of the modulus k.

670 Appendix B

(1945), but programs to evaluate elliptic functions on a computer are notdifficult to write, and some are available commercially.

Fig. B.2. Graphs of the elliptic functions sn x, cn x, dn x for

For applications we need some systematic procedures for reducing equa-tions to the standard form (B.2). Consider the equation

where A, B, C and D are given scalar constants. This can be reduced tostandard form by the change of variables

and a, b, are constants. To perform the reduction and determine theconstants, we differentiate (B.13) to get

The left side of this equation can be expressed in terms of by substituting(B.12) and (B.13), while the right side can be expressed in terms of byusing

Then, by equating coefficients of like powers in y, we obtain

When these four equations are solved for the four unknowns b, a, andthe solution to (B.12) is given explicitly by (B.13). To prepare for this, weeliminate from the third equation and from the second, putting theequations in the form

Elliptical Functions 671

After the cubic equation (B.16a) has been solved for b, the quadraticequation (B.16b) can be solved for a. Then the values of a and b can be usedto evaluate and

The theory of elliptic functions is rich and complex, a powerful tool formathematical physics. We have discussed only some simpler aspects of theoryneeded for applications in the text.

Exercises

(B.1) Establish the derivatives

where(B.3) Find a change of variables that transforms

into an equation of the form (B.12).

(B.2) Show that and are solutions of the differentialequations

Appendix C

Tables of Units, Constants and Data

C-1. Units and Conversion FactorsLength

1 kilometer (km)1 angström (Å)1 fermi1 light-year1 astronomical unit (AU)

Time, frequency1 sidereal day

1 mean solar day1 sidereal year1 sidereal month1 Hertz

Force, Energy, Power1 newton (N)1 joule (J)

1 MeV1 watt

Magnetic Field1 tesla

C-2. Physical ConstantsGravitational constantSpeed of lightElectron massProton mass

672

Tables of Units, Constants and Data 673

Neutron massElectron charge

C-3. The EarthMassEquatorial radiusPolar radiusFlatteningPrincipal moments of inertia

PolarEquitorial

Inclination of equatorLength of year (Julian)

C-4. The SunSolar massSolar radiusSolar luminosityMean Earth-Sun distance

C-5. The MoonLunar massLunar radiusInclination of lunar equator

to eclipticto orbit

Mean Earth-Moon distanceEccentricity of orbitSidereal period

C-6. The Planets

Hints and Solutions for SelectedExercises

“An expert is someone who has made all the mistakes”H. Bethe

“Therefore we should strive to make mistakes as fast aspossible”

J. Wheeler

Section 1-7.(7.1c) (Given)

(Addition Property)(Associativity)(Additive Inverse)(Additive Identity)

(7.1d)

(7.2a)

This is undefined if

(7.2b) If then is idempotent.

(1.1)

Note that the ambiguity in writing is inconsequential.(1.2) If and then

674

(7.2c) If then So if thenwhich is a contradiction.

(7.2d) If and then Hence

Section 2-1.

Hints and Solutions for Selected Exercises 675

soand then, from Exercise (1.1),

(1.3) provided the denominator does not vanish.

(1.4) First show that Then use to get

(1.5)

(1.6)Separate scalar and bivector parts.

(1.7)

(1.8)Repeat the operation until a has been moved to the far right withinthe bracket. Then use which follows fromExercise (2.4).

(1.11) One proof uses equations (1.8) and (1.14).

(1.12)

(1.13)

(1.14) Use (1.24a, b) and Exercise (1.13) as follows,

(1.15) If then there exist scalars such thatHence

If then

(1.17)

we would have and which isinconsistent with

676 Hints and Solutions for Selected Exercises

Section 2-3.

(3. 1a) By (1.5),or by (1.4),

(3.1b) a vector.

impliesand also

(3.2)

(3.3)

Equate vector and trivector parts separately to get the first twoidentities.

Equate vector and trivector parts separately.

(3.4)

(3.5)

(3.7) Make the identifications Hamilton chose alefthanded basis, in contrast to our choice of a righthanded basis.

(3.8)

The last step follows from

(3.9)

(3.10) This problem is the same as Exercise (1.4),


Section 2-4.

(4.5b)

(4.7) The figure is a regular hexagon with external anglethere are two other

(4.8a)For this reduces to

(4.8b, c) These identities were proved in Exercise 1.1. Note that the secondidentity admits the simplification if

For and the identities reduce to

(4.8d, e) Note that because Thus,

The desired identities are obtained by adding and subtracting thisfrom the identity

For and these identities reduce to

(4.9b) Eliminateand

(4.10)(4.11a) Establish and interpret the identity

Note that if any two terms in the identity vanish, the third vanishesalso. Note that this is an instance of logical transitivity, and that thetransitivity breaks the symmetry of the relation.

(4.11b) Establish, interpret and use the identity


Alternatively, one can argue that andimplies Here the transitivity in the argument

is quite explicit.(4.11c) Use the facts that and

Whence,What more is needed to conclude that

Section 2-5.

(5.8)

The quantity has the form of a complex number and, as canbe verified by long division, submits to the binomial expansion

which converges for

Section 2-6.

(6.2) (a) Equation (6.2) implies(6.2) (b) {x} = half line with the direction u and endpoint a.

(6.3)

(6.4)(6.5) The solution set is the line of intersection of the A-plane with the

B-plane.

(6.6)

(6.7)

Hence,

But so

(6.1) etc.


(6.8)(6.10) Comparing Figure 6.2 with Figure 6.4b, we see that we can use

(6.17) to get

(6.11) The area of the quadrilateral 0, a, b, c is divided into four parts bythe diagonals. The theorem can be proved by expressing the div-ision ratios in terms of these four area.

(6.12) An immediate consequence of Equation (6.13).(6.13) By Equation (6.12), we may write

Whence,

So, if

If the lines are parallel and may be regarded as in-tersecting at After deriving similar expressions for p and q wecan show that

and Exercise (6.12) can be used again.(6.14) implies(6.15) or, to use the special form of Exercise

(6.14),(6.16) Expand(6.17) At the points of intersection(6.18) The equation for the line tangent to the circle at the point

can be written Its square isEvaluate at and solve for d.


(6.19)

(6.21a)

(6.22)

(6.23) (a) Write ThenHence, and which describe a

hyperbola in terms of rectangular coordinates.(b) A circle with radius and center at(c) The evolvents of the unit circle, i.e. the path traced by a point

on a taut string being unwound from around the unit circle.(d) A lemniscate. Note that it can be obtained from the hyperbola

in (a) by the inversion(6.24) (a) For this describes a cone with vertex at the

origin and vertex angle given by cos it reducesto a line when and a plane when Only zero is asolution when

(b) Interior of a half cone for(c) Cone with vertex angle symmetry axis and plane of the

cone.(d) describes a hyperboloid asymp-

totic to the cone in (c).(e) Paraboloid.(f) For and circle if ellipse if

parabola if hyperbola if

Section 2-7.

(7.1)

(7.2b) Differentiate(7.3) Generate the exponential series by a Taylor expansion about

and write Conversely, differentiate the exponential seriesto get F.

(7.4) (a) Use the fact that the square of a k-blade is a scalar,(b) Consider where A is a constant bivector.

(7.5) Separate d/dt into scalar and bivector parts.


(7.7) and according to Equation (7.11),

Section 2-8.

(8.2a)

(8.2b)

(8.2f)

(8.5) Write and

Then

and

So

and

This has the scalar part

and the bivector part

The principal value is obtained by integrating along the straight linefrom a to b or along any curve in the plane which can be con-tinuously deformed into that line without passing through theorigin. If the straight line itself passes through the origin, thebivector part of the principal part can be assigned either of the


values If the curve winds about the origin k times, the valueof the integral differs from the principal part by the amountwith the positive (negative) sign for counterclockwise (clockwisewinding.

Section 3-2.

(2.2) This expression has the draw-back that r and are not independent variables. For given and r,the two roots are times of flight to the same point by differentpaths. For given they are times of flight to two distinct points onthe same path equidistant from the vertical maximum.

(2.4) Therefore, horizontalrange x is a maximum for fixed and y when

(2.6) Suggestion: Use the Jacobi identity for g, v, r and the fact that thevectors are coplanar.

(2.7)

Section 3-5.

(5.1a)b)c)

(5.2) 11 m(5.3) 120 m/s.(5.5) The heavy ball beats the light one by 2.2 m and 1/20 sec.

Section 3-7.

(7.2)

Section 3-8.

(8.5)where

and the so-called Larmor frequency is defined by

The general solution is


where

The solution can be interpreted as an ellipse with period processing(retrograde) with angular velocity while it vibrates along the B directionwith frequency and amplitude

Section 4-2.

(2.4)

(2.5)

(2.6)

(2.7)

(2.9)

Section 4-3.

(3.5)

(3.6)

(3.9)(3.10)

(3.11)

Section 4-5.

(5.1) for all values of r implies


(5.2) In each case the integral can be simplified by changing to thevariable or to

(5.3) Investigate derivatives of the effective potential higher than secondorder.

Section 4-7.

(7.4)

(7.5)

(7.6)

Section 5-1.

(1.1)(1.2)

(1.3)

(1.4) only if all vanish provided

(1.5)(1.6) Solution from Exercise (2–1.3)

(1.7) Solution from Exercise (2–1.4)

(1.8) Use Equations (2–1.16) and (2–1.18).(1.10)


(1.12)

Section 5-2.

(2.1)

(2.3b)(2.5)

(2.6b) .(2.7) A quadric surface centered at the point a, as described in Exercise

(2.6) with No solution if all eigenvalues of are negative.

Section 5-3.

(3.1)(3.2)(3.3)(3.5)(3.6) Use the relations and(3.7) From Exercise (3.6)

(3.12)(3.13)

(3.15)


(3.16) Eigenvector Principal valuesPrincipal vectors

whence tan

Section 5-4.

(4.2)

(4.6)(4.7)(4.8)

Section 5-5.

(5.3) Using Equations (3.42a, b, c),

(5.7)

Section 5-6.

(6.1)


East.(b) In an inertial frame, the Easternly velocity of the ball when it is

released is greater than that of the Earth’s surface.South.

Section 6-1.

(1.1) Consider the universe.(1.3) Energy dissipated =

Section 6-2.

Section 6-3.

Section 6-4.

Section 7-2.

with unnormalizedwhere and are nor-

malization factors.


from the center of the sphere.

(2.23) Use the method at the end of Section 5–2.

Section 7-6.


Appendix A

= 2ab sin C + 2bc sin A + 2ca sin A.

From Exercise (A.6),

References

There are many fine textbooks on classical mechanics, but only a couple are mentionedbelow as supplements to the present text. Most students spend too much time studyingtextbooks. They should begin to familiarize themselves with the wider scientific literatureas soon as possible. The sooner a student penetrates the specialist literature on topics thatinterest him, the more rapidly he will approach the research frontier. He should not beafraid to tackle advanced monographs, for he will find that they often contain more lucidtreatments of the basics than introductory texts, and the difficult parts will alert him tospecifics in his background that need to be filled in. Rather than read aimlessly in a broadfield, he should focus on specific topics, search out the relevant literature, and determinewhat is required to master them. Above all, he should learn to see the scientific literatureas a vast lode of exciting ideas which he can mine at will by himself.

Most of the references below are intended as entries to the literature on offshoots andapplications of mechanics. Many are classics in their fields, and some are advanced mono-graphs, but all of them will yield rich rewards to the dedicated student. This is just asampling of the literature with no attempt at completeness on any topic.

Supplementary Texts

The books by French and Feynman are notable for their rich physical insight communicatedwith a minimum of mathematics. Although both are introductory textbooks, they can beread with profit by advanced students. French’s book is especially valuable for its historicalinformation, which should be part of every physicists education.

A. P. French, Newtonian Mechanics, Norton, N.Y. (1971).R. P. Feynman, Lectures on Physics, Vol. 1. Addison-Wesley, Reading (1963).

References on Geometric Algebra

The present book is the most complete introduction to geometric algebra (GA) and itsapplications to mechanics. The book edited by Baylis surveys GA applications to manyother domains of physics, engineering and mathematics, though there are serious differencesin viewpoint among the authors, and most chapters are pitched at the graduate level. Theelectrodynamics textbook by Jancewicz is pitched at the intermediate level. The othertwo books are advanced monographs. Spacetime Algebra deals tersely with applications torelativity, and it articulates smoothly with Chapter 9 of this book. The book by Hestenesand Sobczyk is devoted exclusively to mathematical developments at an advanced level.

W. E. Baylis (Ed.) Clifford (Geometric) Algebras with Applications in Physics, Mathe-matics, and Engineering. Birkhaüser, Boston (1996).

D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus, a Unified Languagefor Mathematics and Physics. D. Reidel, Dordrecht (1984). [Referred to as GeometricCalculus in the text].

D. Hestenes, Space-Time Algebra, Gordon and Breach, N .Y. (1966).B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics, World Scientific, Sin-

gapore (1989).

690

References 691

Chapter 1

A satisfactory history of geometric algebra has not yet been written. But Kline tracesthe interplay between geometry and algebra, mathematics and physics in their historicaldevelopment. The scholarly work by Van der Waarden shows clearly the common historicalorigins of geometry and algebra. Clifford’s book is one of the best popular expositions everwritten on the role of mathematics in science. Grassmann’s impressive contributions arediscussed in Schubring’s book.

M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford U. Press, N.Y.(1972).

B. L. Van der Waarden, Science Awakening, Wiley, N.Y. (1963).W. K. Clifford. Common Sense of the Exact Sciences (1978), reprinted by Dover. N.Y.

(1946).G. Schubring (Ed.), Hermann Günther Grassmann (1809–1877) —Visionary Scientist

and Neohumanist Scholar, Kluwer: Dordrecht (1997).

Section 2-6

Zwikker gives an extensive treatment of plane analytic geometry, using complex numbersin a manner closely related to the techniques of Geometric Algebra.

C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, N.Y.(1963).

Section 3-1.

This magnificently edited and annotated collection of Newton’s papers provides valuableinsight into Newton’s genius. One can see, for instance, the extensive mathematical prepa-ration in analytic geometry that preceded his great work in mechanics.

D. T. Whiteside (Ed.). The Mathematical Papers of Isaac Newton, Cambridge U. Press,Cambridge (1967–81), 8 Vol.

Section 3-5.

The forces of fluids on moving objects are extensively analyzed theoretically and empiricallyin Batchelor’s classic.

G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge U. Press, N.Y. (1967).

Section 3-6 through 3-9.

Feynman gives a good introduction to electromagnetic fields and forces.

R. P. Feynman, The Feynman Lectures on Physics, Vol. II, Addison-Wesley, Reading(1964).

For a more advance, but more compact, treatment using geometric algebra, see the booksby Baylis, Jancewicz and Hestenes above.

692 References

Section 5-3, 4.

The spinor approach to rotations is applied to biomechanics by Hestenes below and torobotics and computer vision by Lasenby in the above book edited by Baylis.

D. Hestenes, Invariant Body Kinematics: I. Saccadic and compensatory eye movements.II. Reaching and Neurogeometry. Neural Networks 7; 65–77, 79–88 (1994).

Section 6-3.

Brillouin’s classic is an object lesson in how much can be accomplished with a minimumof mathematics. He discusses electrical-mechanical analogies as well as waves in crystals.

L. Brillouin, Wave Propagation in Periodic Structures, McGraw-Hill, N.Y. 1946 (Dover.N.Y. 1953).

Physics students will do well to sample the vast engineering and applied mathematicsliterature on linear systems theory.

Section 6-4.

Herzberg is still one of the most important references on molecular vibrations. Califanogives a more up-to-date treatment of group theoretic methods to account for molecularsymmetry. Further improvement in these methods may be expected from employment ofgeometric algebra.

G. Herzberg, Molecular Spectra and Molecular Structure, II. Infrared and Raman Spectraof Polyatomic Molecules, D. Von Nostrand Co., London, (1945).

S. Califano, Vibrational States, Wiley, N.Y. (1976).

Section 6-5.

The most extensive survey of work on the three body problem is

V. Szebehely, Theory of Orbits, Academic Press, N.Y. (1967).

Section 7-4.

This is one of the standard advanced references on the theory of spinning bodies, as wellas the three body problem:

E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies.Cambridge. 4th Ed. (1937).

Chapter 8

Stacey and Kaula present fine introductions to the rich field of geophysics and its gener-alization to planetary physics, showing connections to celestial mechanics. The book byMunk and MacDonald is a classic on the Earth’s rotation.

Roy gives an up-to-date introduction to celestial mechanics and astromechanics com-bined. Kaplan gives a more complete treatment of spacecraft physics.

References 693

F. D. Stacey, Physics of the Earth, Wiley, N.Y. (1969).W. M. Kaula, An Introduction to Planetary Physics, Wiley, N.Y. (1968).W. Munk and G. MacDonald, The Rotation of the Earth, Cambridge U . Press, London

(1960).A. E. Roy, Orbital Motion, Adam Hilger, Bristol, 2nd Ed. (1982).M. H. Kaplan, Modern Spacecraft Dynamics and Control. Wiley, N.Y. (1976).

Section 8-3.

Newton’s lunar theory was to be the crowning glory of his Principia. The story of hisfrustration and failure to match his calculations to the available data is told by the dis-tinguished Newton scholar D.T. Whiteside. A modern account of the elaborate standardcalculations is given by Brouwer and Clemence.

D.T. Whiteside, Newton’s Lunar Theory: From High Hope to Disenchantment, Vistas inAstronomy 19: 317–328 (1976).

D. Brouwer and G.M. Clemence, Methods of Celestial Mechanics, Academic Press, London(1961).

Section 8-4.

Steifel and Schiefele is the only book on applications of the KS equation. It providesmany important insights into computational theory and technique. Vrbik has pursued theapplication of geometric algebra (quaternions) to perturbations in celestial mechanics.

E. L. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics, Springer-Verlag.N.Y. (1971).

J. Vrbik, Celestial mechanics via quaternions, Can. J. Physics, 72, 141–146 (1994).Two-body perturbed problem revisited, Can. J. Physics, 73, 193–198 (1995).Perturbed Kepler problem in quaternionic form, J. Phys. A, 28, 6245–6252 (1995).Resonance formation of Kirkwood gaps and asteroid clusters, J. Phys. A, 29,

3311–3316 (1996).Oblateness perturbations to fourth order, Mon. Not. R. Astron. Soc., 291, 65–70

(1997).

Chapter 9.

The first of these articles explains how the geometric algebra in this book relates to the moregeneral and powerful spacetime algebra (STA). The second article shows how the generalmotion in uniform fields (Sections 9-3 and 9-5) can be treated with STA and extended toplane wave and Coulomb fields.

D. Hestenes, Proper Particle Mechanics, J. Math. Phys. 15 1768–1777 (1974).Proper Dynamics of a Rigid Point Particle, J. Math. Phys. 15 1778–1786 (1974).

Appendix A

Jahnke and Emde is a standard reference on elliptic functions and elliptic integrals.

E. Jahnke and F. Emde, Tables of Functions, Dover, N.Y. (1945).

Index

acceleration, 98, 312centripetal, 312Coriolis, 312

ambient velocity, 146amplitude of an oscillation, 168analyticity principle, 122angle, 66

radian measure of, 219angular momentum 195ff

base point, 423bivector, 196change of, 423conservation, 196, 338induced, 330internal, 337intrinsic, 424, 655orbital, 337total, 337vector, 196

Angular Momentum Theorem, 338anharmonic oscillator, 165anomaly,

eccentric, 532mean, 532true, 532, 573

apocenter, 213apse (see turning point),area,

directed, 70integral, 112ff, 196

associative rule, 27, 32, 35astromechanics, 512asymptotic region, 210, 236attitude, 420

element, 529, 549spinor, 420

Atwood’s machine, 354axode, 428

ballistic trajectory, 215

barycentric coordinates, 82basis, 53

of a linear space, 53, 363multivector, 53vectorial, 49, 260

beats, 365Big Bang, 611billiards, 498, 503binding energy, 638bivector (2-vector), 21basis of, 56

codirectional, 24interpretations of, 49

blade, 34boost, 581, 586, 605Brillouin zone, 374

Cayley-Klein parameters, 480, 485, 495celestial mechanics, 512ffcelestial pole, 458, 538celestial sphere, 466, 537center of gravity, 433center of mass, 230, 336

additivity principles for, 437of continuous body, 434symmetry principles for, 435ff

(see centroid)system, 644

center of mass theorem, 336centroid, 438chain rule, 100, 105, 108Chandler wobble, 458characteristic equation, 166, 171, 383Chasles’ theorem, 305chord, 79circle, equations for, 87ffClassical Field Theory, 514Clifford, 59coefficient of restitution, 505, 511collision,

694

Index 695

elastic, 236, 505inelastic, 346, 505

commutative rule, 15, 35commutator, 44Compton effect, 643Compton wavelength, 644configuration space, 351, 382congruence, 3, 303, 605conjugate, 579, 580conjugation, 580conicoid, 91conic (section), 90ff, 207constants of motion,

for Lagrange problem, 476for rigid motion, 425for three body problems, 399

constraint, 181bilateral, 188for rolling contact, 492for slipping, 497holonomic, 185, 351, 354unilateral, 188

continuity, 97coordinates,

complex, 371ecliptic, 238ffequitorial, 238ffgeneralized, 350, 381ignorable (cyclic), 358Jacobi, 406mass-weighted, 385normal (characteristic), 362, 364polar, 132, 194rectangular, 132symmetry, 388

couple, 430covariance, 619covariant kinematics, 615Cramer’s rule, 254cross product, 60

degrees of freedom, 351derivative,

by a vector, 117convective, 109directional, 105, 107of a spinor, 307partial, 108scalar, 98total, 109

Descartes, 5determinant, 62, 255

of a frame, 261of a linear operator, 255, 260of a matrix, 258, 260

differential, 107exact, 116

differential equation, 125dihedral angle, 662dilation, 13, 52dimension, 34, 54Diophantes, 9directance, 82, 87, 93, 427direction, 11

of a line, 48of a plane, 49

dispersion relation, 373displacement,

rigid, 303, 305screw, 305

distance, 79distributive rule, 18, 25, 31, 35Doppler effect, 607

aberration, 609, 644drag, 146

atmospheric, 215, 563pressure, 149viscous, 149(see force law)

drag coefficient CD, 147drift velocity, 159dual, 56, 63dynamical equations, 454

(see equations of motion),dynamics, 198

eccentricity, 90, 205eccentricity vector, 91, 205, 527ecliptic, 466, 539eigenvalue problem, 264ff

brute force method, 384eigenvalues, 264ff

degenerate, 266eigenvectors, 264ff, 272Einstein, 574elastic modulus, 374elastic scattering, 646elastic solid, 360electromagnetic wave, 174elementary particles, 638

696 Index

ellipsoid, 276ellipse, 91, 96, 173, 174, 199, 203, 208

semi-major axis of, 212elliptic functions, 222, 478, 481, 668ff

modulus of, 668elliptic integral, 482, 490, 545, 547

B-669ffenergy,

conservation, 170Coriolis, 342diagrams, 223, 229dissipation, 177, 241ellipsoid, 487internal, 342ffkinetic, 182, 337

internal, 341rotational, 338translational, 337

potential, 182storage, 177, 364total, 182, 206, 528transfer, 238, 344, 364vibrational, 342

energy-momentum conservation, 633epicycle, 201epitrochoid, 201, 204equality, 12, 37equations of motion, 125

rotational (see spinor equations),340, 420

secular, 531for orbital elements, 531

translational, 335, 420equiangular spiral, ISSequilibrium, 379

mechanical, 429point, 409

equimomental rigid bodies, 448equinoxes, 539equipotential surface, 116, 185escape velocity, 214Euclid, 29Euclidean group, 607Euclidean spaces

2-dimensional, 543-dimensional, 54n-dimensional, 80

Euler, 121Euler angles, 289, 294, 486, 490, 538

Euler’s Law (equation), 340, 420, 454components of, 422ff

Euler parameters, 382, 315event, 584event horizon, 623exponential function, 66, 73ff, 281

factorization, 45Faraday effect, 179Fermi-Walker transport, 652field, 104first law of thermodynamics, 344fluid resistance, 146ffforce, 121

4-force, 618binding, 164body, 125centrifugal, 318, 332conservative, 181, 219contact, 125Coriolis, 319, 322, 324, 328electromagnetic, 620fictitious (see force law), 317generalized, 353impulsive, 214, 501perturbing, 143, 165, 527superposition, 122tidal, 520

force constants, 380force field, 184

central, 219conservative, 184, 219

force law, 122conservative, 181ffconstant, 126Coulomb, 205,

with cutoff, 251electromagnetic, 123, 155, 620frictional, 192, 471gravitational, 123, 200, 205, 513Hooke’s, 122, 361, 364inverse square, 200magnetic, 151phenomenological, 195resistive (see drag), 146

linear, 134, 154quadratic, 140

forces on a rigid body,

Index 697

concurrent, 433equipollent, 428parallel, 429reduction of, 428

frame (see basis), 261body, 339Kepler, 529reciprocal, 262

frequency,cutoff, 371cyclotron, 154Larmor, 328normal (charcteristic), 362

degenerate, 362oscillator, 168resonant, 177

multiple, 397

Galilean transformation, 574, 583geoid, 524, 526geometric algebra, 53, 55, 80geometric product, 31, 39geometry,

analytic, 78ffcoordinate, 78Euclidean, 79non-Euclidean, 79

geopotential, 525Gibbs, 60golden ratio, 226grade, 22, 30, 34gradient, 116Grassmann, 12, 14, 28gravitational field, 513

force exerted by, 513, 520of an axisymmetric body, 518of an extended object, 515ffsource, 513superposition, 514

gravitational potential, 514harmonic (multipole) expansion of, 517of a spherically symmetric body, 516

gravitational quadrupole tensor, 517, 542gravity assist, 239, 242group,

abstract, 296continuous, 298dirotation, 296

orthogonal, 2 99representation, 297rotation, 296ffsubgroup of, 299, 306transformation, 295translation, 300ff

guiding center, 158gyroscope, 454gyroscopic stiffness, 455ff

Hall effect, 160Halley’s comet, 214Hamilton, 59, 286Hamilton's theorem, 295harmonic approximation, 380harmonic oscillator, 165

anisotropic, 168coupled, 361damped, 170forced, 174in a uniform field, 173, 202, 325isotropic, 165

headlight effect, 611heat transfer, 345helix, 154Hill’s regions, 416history, 578hodograph, 127, 204Hooke’s law, 122, 166hyperbola, 91, 96, 208

branches, 213hyperbolic functions, 74hyperbolic motion, 623hypotrochoid, 202, 204

idempotent, 38impact parameter, 211, 245impulse, 501impulsive motion, 501inertia tensor, 253, 339, 421, 439inertial frame, 586, 594

additivity principles for, 442calculation of, 439canonical form for, 451derivative of, 340matrix elements of, 445of a plane lamina, 274principal axes of, 422principal values of, 422

Euclidean, 301ffGalilean, 313

698 Index

symmetries of, 448inertial system, 578, 630initial conditions, 125initial value problem, 208

gravitational, 513inner product, 16ff, 33, 36, 39

, 43isotropic,

space, 591spacetime, 591

integrating factor, 134, 139, 152, 173interaction, 121interval, 589, 606

lightlike, 589spacelike, 589squared, 589timelike, 589

invariant, 606inverse, 35, 37inversion, 293, 437

Jacobi identity, 47, 83Jacobi’s integral, 408

Kepler, 200Kepler motion, 527Kepler problem, 2042-body effects on, 233Kepler’s equation, 216ff, 533Kepler’s Laws,

first, 198second, 196, 198third, 197, 198, 200, 203

modification of, 233kinematical equation,

for rotational motion, 454kinematics, 198kinetic energy,

relativistic, 618KS equation, 569

laboratory system, 644Lagrange points, 409Lagrangian, 353Lagrange’s equation, 190, 353, 380Lagrange’s method, 354Lame’s equation, 491Laplace expansion, 43, 261Laplace vector (see eccentricity vector),Larmor’s theorem, 328

lattice constant, 367law of cosines, 19, 69

spherical, 523, 664, 665law of sines, 26, 70

spherical, 665law of tangents, 294lemniscate, 204lever, law of, 430lightcone, 591line,

equations for, 48, 81fflightlike, 589moment of, 82spacelike, 589timelike, 589

line integral, 109ff, 115line vector, 428linear algebra, 254linear dependence, 47linear function, 107, 252ff, 578

(see linear operators)linear independence, 53linear operators, 253ff

adjoint (transpose), 254canonical forms, 263, 270, 282derivative of, 316determinant of, 255, 260inverse, 260

matrix element, 262matrix element, 257matrix representation of, 257nonsingular, 256orthogonal, 277

improper, 278proper, 278

polar decomposition, 291product, 253secular equation for, 265

complex roots, 268degenerate, 266

shear, 295skewsymmetric, 263symmetric (self-adjoint), 263, 269ff

spectral form, 270square root, 271

trace, 295linear space, 53

dimension of, 54linear transformation (see linear operator)Lissajous figure, 169logarithms, 75ff

Index 699

Lorentz contraction, 599Lorentz electron theory, 179Lorentz force, 123Lorentz group, 587Lorentz transformation, 580, 586, 647

active, 605passive, 605

Mach number, 149magnetic spin resonance, 473ffmagnetron, 202magnitude, 3, 6

of a bivector, 24of a multivector, 46of a vector, 12

many body problem, 398constants of motion, 399

mass, 230density, 434reduced, 230total, 336, 434

mass-energy equivalence, 634matrix, 257

determinant of, 259, 260equation, 258identity, 258product, 258sum, 258

mean motion, 532measurement, 2Minkowski, 577model, 378modulus,

of a complex number, 51of an elliptic function, 668of a multivector, 46

Mohr’s algorithm, 273moment arm, 428momentum, 236

4-momentum, 634conservation, 236, 336flux, 347transfer, 238, 240

Mössbauer effect, 643motion, 121

in rotating systems, 317ffrigid, 306fftranslational, 335

(see rotational motion, periodicmotion)

multivector, 34even, 41homogeneous, 41, 12k-vector part of, 34 39odd, 41reverse, 45

natural frequency, 168Newton, 1, 120, 124Newton’s Law of Gravitation, 398

universality of (see force law), 201, 203Newton’s Laws of Motion,

zeroth, 615first, 579, 615second, 41, 615, 618third, 335

in relativity, 633nodes,

ascending, 538line of, 290precession of, 540

nonrelativistic limit, 584normal (to a surface), 116normal modes, 362

degenerate, 383expansion, 364nondegenerate, 383normalization, 377orthogonality, 369wave form, 369

number, 3, 5complex, 57directed, 11, 12, 34imaginary, 51real, 10, 11, 12

nutation,luni-solar, 551of a Kepler orbit, 540of a top, 470of Moon’s orbit, 550

oblateness,constant J2, 518of Earth, 459, 467perturbation, 542, 560

Ohm’s law, 137operational definition, 576operators (see linear operators), 50orbit, 121orbital averages, 253ff

700 Index

orbital elements, 527Eulerian, 538secular equations for, 531

orbital transfer, 214orientation, 16, 23, 51origin, 79oscillations,

damped, 393forced, 395free, 382phase of, 168small (see vibrations), 378

osculating orbit, 528outer product, 20, 23, 36, 39

of blades, 43outermorphism, 255

parabola, 91, 96, 126, 207Parallel Axis Theorem, 424parenthesis, 42

preference convention, 42for sets, 48

particle, 121decay, 640unstable, 242

pendulum,compound, 463, 477, 489conical, 467double, 355, 386

damped, 395Foucault, 223gyroscopic, 462simple, 191, 463small oscillations of, 462spherical, 475

pericenter, 91, 213perigee, 213perihelion, 213period,

of an oscillator, 168of central force motion, 221of the Moon, 203

periodic motion, 168, 478perturbation,

oblateness, 542, 560theory, 141, 320gravitational, 527, 541third body, 541

photon, 578, 641physical space, 80

Planck’s law, 642plane,

equations for, 86ffspacelike, 591timelike, 591

Poincaré, 399, 416Poincaré group, 607Poinsot’s construction, 487point of division, 84, 430polygonal approximation, 143position, 80, 121, 420

spinor, 564position space, 314potential, 116

attractive, 229barrier, 228central, 220centrifugal, 224effective, 220, 408, 414gravitational, 514, 516screened Coulomb, 224secular, 545Yukawa, 224

precession, 222luni-solar, 552of Mercury’s perihelion, 542ffof pericenter, 53Uof the equinoxes, 46B, 553relativistic,

General, 559Special, 562

precession of a rigid body,Eulerian free, 456, 467, 475relativistic, 650ffsteady, 463, 483

deviations from, 467Thomas, 654

principal moments of inertia, 446principal values, 269, 292principal vectors, 269, 292

of inertia tensor, 422Principle of Relativity, 574projectile,

Coriolis deflection, 321ffrange, 127, 136terminal velocity, 135time of flight, 130, 132

projection, 16, 65, 270, 661proper mass, 617proper time, 590, 602

Index 701

pseudoscalar,dextral (right handed), 55of a plane, 49, 53of 3-space, 54, 57

pseudosphere, 614

quantity, 34quaternion, 58, 62

theory of rotations, 286

radius of gyration, 447, 463rapidity, 583, 587reaction energy, 639, 645reference frame (body), 314reference system, 317

geocentric, 317heliocentric, 317inertial, 311topocentric, 317

motion of, 327reflection, 278ff

law of, 280rejection, 65regularization, 571relative mass, 617relativity, 575, 607

General Theory, 514, 542, 557, 577of distant simultaneity, 576Special Theory, 562, 577

relaxation time, 135resonance, 176

cyclotron, 162electromagnetic, 175magnetic, 331, 473ffmultiple, 396spin-orbit, 554

rest mass, 617reversion, 45Reynolds Number, 147rigid body classification, 448

asymmetric, 448axially symmetric, 448, 454centrosymmetric, 448

rocket propulsion, 348Rodrigues’ formula, 293rolling motion, 492ffrotation, 50, 278, 280ff

axis, 304canonical form, 282, 288composition, 283group, 295ff, 587

matrix representation, 296spin representation, 296

matrix elements, 286,with Euler angles, 294

oriented, 283parametric form, 282physical, 297right hand rule, 282spinor theory of, 286rotational motion, 317

integrable cases (see spinning top) , 476of a particle system, 338of asymmetric body, 482of the Earth, 327, 551

stability of, 488

satellite,orbital precession, 544perturbation of, 547synchronous orbit, 203

scalar, 12scalar integration, 100scalar multiplication, 12, 24, 31, 35scattering,

angle, 210, 245, 646in CM system, 237, 242in LAB system, 239, 242

Coulomb, 247, 250cross section, 243ff

LAB and CM, 248ffRutherford, 247

elastic, 236for inverse square force, 210ffhard sphere, 246

semi-latus rectum, 91, 212sense (or orientation), 51siderial day, 458simultaneity, 576, 594solar wind, 563solid angle, 244spacetime, 577

homogeneous, 607isotropic, 607map, 584, 589

spatial rotation, 581, 586speed of light, 575sphere, equations for, 87ffspherical excess, 667spin precession,

relativistic, 658

702 Index

spinning top,fast, 466hanging (see precession, rotational

motion), 466Lagrange problem for, 462, 479, 490relativistic, 650rising, 473sleeping, 473slow, 466spherical, 460symmetrical, 454ff

Eulerian motion of, 460reduction of, 459

spinor, 51, 52, 67derivative of, 307Eulerian form, 284improper, 300mechanics, 564parametrizations, 286unitary (unimodular), 280

spinor equation of motion,for a particle, 569for a spherical top, 461relativistic, 657

stability, 165, 227, 380of circular orbits, 228of Lagrange points, 410ffof rotational motion, 488of satellite attitude, 553

state variables, 126Stokes’ Law, 147summation convention, 63super-ball, 507, 510superposition principle,

for fields, 514for forces, 122for vibrations, 363

symmetry of a body, 435ff, 441synchronizing clocks, 594system,

2-particle, 230ffclosed, 346configuration of, 350Earth-Moon, 234harmonic, 382isolated, 232, 336linear, 378many-particle, 334ffopen, 346

systems theory,linear, 378

Taylor expansion, 102, 107, 164temperature, 346

tensor, 253three body problem, 400

circular restricted, 407periodic solutions, 416

classification of solutions, 404collinear solutions, 402restricted, 406triangular solutions, 402

tidal friction, 522tides, 522time dilation, 596tippie-top, 476torque, 338

base point, 424gravitational, 520moment arm, 428

translation, 300(see group)

trivector, 26trigonometric functions, 74, 281trigonometry, 20, 68

identities, 71, 294spherical, 661

trochoid, 159, 217turning points, 213, 221, 227twin paradox, 600

units, 672

variables,interaction, 334kinematic, 421macroscopic, 345object, 421position, 334state, 420

vector, 12addition, 15axial, 61collinear (codirectional), 16, 64identities, 62negative, 15orthogonal, 49, 64orthonormal, 55polar, 61rectangular components, 49, 56square, 35units, 13

vector field, 184vector space, 49, 53velocity, 98

4-velocity, 6156-velocity, 651additional theorem, 314

Index 703

angular, 309complex, 427filter, 160rotational with Euler angles, 308, 423,

315, 490spinor, 564translational, 309

vibrations,of H2O, 392lattice, 366molecular, 341, 387ffsmall, 341, 378

Vieta, 9

wave,harmonic, 372polarized, 375standing, 372traveling, 372

wavelength, 369wave number, 369weight,

apparent, 318true, 318

work, 183, 342ffmicroscopic, 345

Work-Energy Theorem, 343wrench, 426

reduction of, 431superposition principle, 428

Zeeman effect, 332zero, 14Zeroth Law, 80, 576

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