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Appendix A Preliminaries on Vectors and Calculus
CAVEAT LECTOR
In writing this primer, I have assumed that the reader has had courses in linear algebra and calculus. This being so, I have more often than not found that these topics have been forgotten. Here, I review some of the basics. But it is a terse review, and I strongly recommend that readers review their own class notes and other texts on these topics in order to fill the gaps in their knowledge.
Students who are able to differentiate vectors, and are familiar with the chain and product rules of calculus, have a distinct advantage in comprehending the material in this primer and in other courses. I have never been able to sufficiently emphasize this point to students at the beginning of an undergraduate dynamics course.
A.l Vector Notation
A fixed (right-handed) Cartesian basis for Euclidean three-space £3 is denoted by the set {Ex, Ey, Ez}· These three vectors are orthonormal (i.e., they each have a unit magnitude and are mutually perpendicular).
For any vector b, one has the representation
b = bxEx + byEy + bzEz ,
where bx, by, and bz are the Cartesian components of the vector b (cf. Figure A.l).
182 Appendix A. Preliminaries on Vectors and Calculus
FIGURE A.l. A vector b
A.2 Dot and Cross Products
The two most commonly used vector products are the dot and cross products. The dot product of any two vectors u and w is a scalar defined by
u · W = UxWx + UyWy + UzWz = llullllwll cos(/),
where "( is the angle subtended by u and w, and lib II denotes the norm (or magnitude) of a vector b:
Clearly, if two vectors are perpendicular to each other, then their dot product is zero.
One can use the dot product to define a unit vector n in the direction of any vector b:
b n = lfbii.
This formula is very useful in establishing expressions for friction forces and normal forces.
The cross product of any two vectors band cis a vector that is perpendicular to the plane they define:
This expression for the cross product can be expressed in another form involving the determinant of a matrix:
[Ex Ey
b X c = -c X b = det bx by Cx Cy
You should notice that if two vectors are parallel, then their cross product is the zero vector 0.
A.3 Differentiation of Vectors 183
A.3 Differentiation of Vectors
Given a vector u, suppose it is a function of time t: u = u(t). One can evaluate its derivative using the product rule:
du dux duy duz dEx dEy dEz dt = dt Ex+ dtEy + dt Ez + Uxdt + Uydt + Uzdt.
However, Ex, Ey, and Ez are constant vectors (i.e., they have constant magnitude and direction). Hence, their time derivatives are zero:
du = duxE duyE duzE dt dt X + dt y + dt z '
We can also use the product rule of calculus to show that
d du dw -(u·w) =- ·w+u·-dt dt dt ,
d du dw dt ( U X W) = dt X W + U X dt .
These results are obtained by representing the vectors u and w with respect to a Cartesian basis, then evaluating the left- and right-hand sides of both equations and showing their equality.
To differentiate any vector-valued function e(8(t)) with respect tot, we use the chain rule:
de_ d8 de_ d8 (dcxE dcyE dczE) dt - dt d8 - dt d8 x + d8 Y + d8 z ·
When differentiating it is important to distinguish a function and its value. For example, suppose 8 = t2 and the function f as a function of time t is f(t) = t. Then, f as a function of 8 is }(8) = 8, but j(8) = 82 -=!= }(8).
A.4 A Ubiquitous Example of Vector Differentiation
One of the main sets of vectors arising in any course on dynamics is { er, eo, ez}:
er cos(B)Ex + sin(B)Ey,
eo -sin( B)Ex + cos( B)Ey ,
ez Ez.
We also refer the reader to Figure A.2. In the above equations, B is a function of time.
Using the previous developments, you should be able to show the following results:
der . de=- sm(B)Ex + cos(B)Ey = e0 ,
184 Appendix A. Preliminaries on Vectors and Calculus
FIGURE A.2. The vectors er and e9
~~ =- cos(O)Ex- sin(O)Ey = -er,
A useful exercise is to evaluate these expressions and graphically represent them for a given O(t). For example, O(t) = 10t2 + 15t.
Finally, you should be able to show that
er X e9 = ez, ez X er = eiJ, eo X ez = er,
er·er=1, eo ·e9 = 1, ez · ez = 1,
er·e9=0, eo. ez = 0, er. ez = 0.
In other words, {en e9, ez} forms an orthonormal set of vectors. Furthermore, since ez · (er x eo)= 1, this set of vectors is also right-handed.
A.5 Ordinary Differential Equations
The main types of differential equations appearing in undergraduate dynamics courses are of the form ii = f(u), where the superposed double dot indicates the second derivative of u with respect to t. The general solution of this differential equation involves two constants: the initial conditions for u (to) = uo, and its velocity it (to) = ito. Often, one chooses time such that to= 0.
The most comprehensive source of mechanics problems that involve differential equations of the form ii = f(u) is Whittaker's classical work [67]. It should also be added that classical works in dynamics placed tremendous emphasis on obtaining analytical solutions to such equations. Recently, the engineering dynamics community has become increasingly aware of possible chaotic solutions. Consequently, the existence of analytical solutions is generally not anticipated. We refer the reader to Moon [40] and Strogatz [62] for further discussions on, and references to, this matter. Further perspectives can be gained by reading the books by Barrow-Green [3] and Diacu and Holmes [20] on Henri Poincare's seminal work on chaos, and Peterson's book [48] on chaos in the solar system.
A.5 Ordinary Differential Equations 185
A.5.1 The Planar Pendulum
One example of the above differential equation arises in the planar pendulum discussed in Chapter 2. Recall that the equation governing the motion of the pendulum was
mLO = -mg cos( 0) .
This equation is of the form discussed above with u = (} and f(u) = -gcos(u)jL. Here, f is a nonlinear function of u. Given the initial conditions(} (to) = 00 and iJ (to) = 00 , this differential equation can be solved analytically. The resulting solution involves special functions that are known as Jacobi's elliptic functions. 1 Alas, these functions are beyond the scope of an undergraduate dynamics class, so instead one normally is required to use the conservation of the total energy E of the particle to solve most posed problems involving this pendulum.
A.5.2 The Projectile Problem
A far easier example arises in the motion of a particle under the influence of a gravitational force -mgEy. There, the differential equations governing the motion of the particle are
mx = 0 , mjj = -mg, mz = 0.
Clearly, each of these three equations is of the form u = f(u). The general solution to the second of these equations is
Y (t) =Yo+ Yo (t- to) - ~ (t- to) 2 •
Here, y (to) =Yo andy (to) = y0 are the initial conditions. You should verify the solution for y(t) given above by first examining whether it satisfies the initial conditions and then seeing whether it satisfies the differential equation jj = -g. By setting g = 0 and changing variables from y to x and z, the solutions to the other two differential equations can be obtained.
A. 5. 3 The Harmonic Oscillator
The most common example of a differential equation in mechanical engineering is found from the harmonic oscillator. Here, a particle of mass m is attached by a linear spring of stiffness K to a fixed point. The variable x is chosen to measure both the displacement of the particle and the displacement of the spring from its unstretched state. The governing differential equation is
mx = -Kx.
1 A discussion of these functions, in addition to the analytical solution of the particle's motion, can be found in Lawden [36], for instance.
186 Appendix A. Preliminaries on Vectors and Calculus
This equation has the general solution
x(t) = xocos ( ~ (t- to)) +xo{:sin ( ~ (t- to)) ,
where x (to) = x0 and x (to) = x0 are the initial conditions.
A.5.4 A Particle in a Whirling Tube
The last example of interest arises in problems concerning a particle of mass m that is in motion in a smooth frictionless tube. The tube is being rotated in a horizontal plane with a constant angular speed n. The differential equation governing the radial motion of the particle is
This equation has the general solution
r(t) = r0 cosh (n (t- t0 )) +~sinh (n (t- t0 )) ,
where r (to) = r 0 and r (to) = r0 are the initial conditions.
Appendix B Weekly Course Content and Notation in Other Texts
Abbreviations
For convenience in this appendix, we shall use the following abbreviations: BF, Bedford and Fowler [6]; BJ, Beer and Johnston [7]; H, Hibbeler [33]; MK, Meriam and Kraige [39]; RS, Riley and Sturges [50]; and S, Shames [56].
B.l Weekly Course Content
The following is an outline for a 15-week (semester-long) course in undergraduate engineering dynamics. Here, we list the weekly topics along with the corresponding sections in this primer. We also indicate the corresponding sections in other texts. This correspondence is, of course, approximate: all of the cited texts have differences in scope and emphasis.
Normally, the course is divided into three parts: a single particle, systems of particles, and (planar dynamics of) rigid bodies. The developments in most texts also cover the material in this order, the exception being Riley and Sturges [50].
188 Appendix B. Weekly Course Content and Notation in Other Texts
Week Topic Primer Other Number Section Texts
1 Single Chapter 1 BF: Ch.1, 2.1-2.3, 3.1-3.4 Particle: BJ: 11.1-11.11, 12.5 Cartesian H: 12.1-12.6, 13.4
Coordinates MK: 1/1-1/7, 2/2, 2/4, 3/4 RS: 13.1-13.4, 15.1-15.3
S: 11.1-11.4, 12.1-12.4
2 Single Chapter 2 BF: 2.3, 3.4 Particle: BJ: 11.14, 12.8
Polar H: 12.8, 13.6 Coordinates MK: 2/6, 3/5
RS: 13.5, 13.7, 15.4 S: 11.6, 12.5
3 Single Chapter 3 BF: 2.3, 3.4 Particle: BJ: 11.13, 12.5
Serret-Frenet H: 12. 7, 13.5 Triads MK: 2/5, 2/7, 3/5
RS: 13.5, 13.7, 15.4 S: 11.5, 12.9
4 Single Chapter 4 BF: 3.4 Particle: BJ: 12.5 Further H: 13.4-13.6 Kinetics MK: 3/5
RS: 15.3, 15.4 S: 12.4, 12.5, 12.9
5 Single Chapter 5 BF: Ch. 4 Particle BJ: 13.1-13.9
Work and H: 14.1, 14.2, 14.4-14.6 Energy MK: 3/6, 3/7
RS: 17.1-17.3, 17.5-17.10 S: 13.1-13.5
6 Single Chapter 6 BF: 5.1, 5.2, 5.4 Particle: Sects. 1 & 2 BJ: 12.2, 12.7, 12.9,13.11
Linear and H: 15.1, 15.5-15.7 Angular MK: 3/9, 3/10
Momentum RS: 19.2, 19.5 S: 14.1, 14.3, 14.6
B.1 Weekly Course Content 189
Week Topic Primer Other Number Section Texts 7 Collisions of Chapter 6 BF: 5.3
Particles Sects. 3-5 BJ: 13.12-13.14 H: 14.3, 14.6, 15.4
MK: 3/12 RS: 19.4
S: 14.4-14.5
8 Systems of Chapter 7 BF: 7.1, 8.1 Particles BJ: 14.1-14.9
H: 13.3, 14.3, 14.6, 15.3 MK: 4/1-4/5
RS: 17.4-17.8, 19.3, 19.5 S: 12.10, 14.2, 14.7,
13.6-13.9
9 Kinematics Chapter 8 BF: 6.1-6.3 of Rigid BJ: 15.1-15.4 Bodies H: 16.1-16.4
MK: 5/1-5/4 RS: 14.1-14.3
S: 15.1-15.5
10 Kinematics Chapter 8 BF: 6.4-6.6 of Rigid BJ: 15.4-15.8, 15.10-15.15 Bodies H: 16.4-16.8
MK: 5/5-5/7 RS: 14.4-14.6 S: 15.5-15.11
11 Planar Chapter 9 BF: 7.2-7.3, App., 9.2 Dynamics of BJ: Ch. 16
Rigid H: 21.1, 21.2, 17.3 Bodies MK: 6/1-6/3, Apps. A & B
RS: 16.2, 16.3, 20.6 S: 16.1-16.4
12 Planar Chapter 9 BF: 7.4 Dynamics of BJ: Ch. 16
Rigid H: 17.4 Bodies MK: 6/4
RS: 16.4 S: 16.5
190 Appendix B. Weekly Course Content and Notation in Other Texts
Week Topic Primer Other Number Section Texts 13 Planar Chapter 9 BF: 8.1-8.3
Dynamics of BJ: Ch. 16 & 17.1-17.7 Rigid H: 17.5 & Ch. 18
Bodies MK: 6/5. 6/6 RS: 16.4 & Ch. 18 S: 16.6, 17.1-17.3
14 Planar Chapter 10 BF: 8.4 Dynamics of BJ: 17.8-17.11
Rigid H: Ch. 19 Bodies MK: 6/8
RS: 20.1-20.5 S: 17.4-17.7
15 Vibrations Not Covered BF: Ch. 10 BJ: Ch. 19
H: Ch. 2 MK: Ch. 8 RS: Ch. 21
S: Ch. 22
B.2 Notation in Other Texts 191
B.2 Notation in Other Texts
Here, we give a brief summary of some of the notational differences between this primer and those used in other texts. In many of the cited texts only plane curves are considered. Consequently, the binormal vector eb is not explicitly mentioned.
Primer Notation
Cartesian {Ex, Ey, Ez} Basis Vectors
Serret-Frenet { et, en, eb} Triad
Linear Momentum G = mv of a Particle
Corotational Basis or Body Fixed Basis
Other Texts
BF: {i,j, k} BJ: {i,j, k} H: {i,j, k}
MK: {i,j,k} RS: {i,j,k}
S: {i,j,k}
BF: {et,en,-} BJ: { et, en, eb} H: { Ut, Un, Ub}
MK: {et,en,-} RS: {et, en,-}
S: { €t, €n, €t X €n}
BF:mv BJ: L = mv
H:mv MK:G
RS: L = mv S:mV
BF: {i,j, k} BJ: {i,j, k} H: {i,j, k}
MK: {i,j,k} RS: {ex, ey, ez}
S: {i,j, k}
References
[1] S. S. Antman, Nonlinear Problems of Elasticity, Springer-Verlag, New York (1995).
[2] V. I. Arnol'd, Mathematical Methods in Classical Mechanics, SpringerVerlag, New York (1978).
[3] J. Barrow-Green, Poincare and the Three-Body Problem, American Mathematical Society, Providence (1997).
[4] M. F. Beatty, "Kinematics of finite rigid body displacements," American Journal of Physics, 34, pp. 949-954 (1966).
[5] M. F. Beatty, Principles of Engineering Mechanics, I. Kinematics -The Geometry of Motion, Plenum Press, New York (1986).
[6] A. Bedford and W. Fowler, Engineering Mechanics - Dynamics, Addison-Wesley, Reading, Massachusetts (1995).
[7] F. P. Beer and E. R. Johnston, Jr., Vector Mechanics for Engineers: Dynamics, Fifth Edition, McGraw-Hill, New York (1988).
[8] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York (1960).
[9] 0. Bottema and B. Roth, Theoretical Kinematics, North-Holland, New York (1979).
194 References
[10] R. M. Brach, Mechanical Impact Dynamics: Rigid Body Collisions, John Wiley & Sons, New York (1991).
[11] J. Casey, "A treatment of rigid body dynamics," ASME Journal of Applied Mechanics, 50, pp. 905-907 (1983) and 51, p. 227 (1984)
[12] J. Casey, Elements of Dynamics, Unpublished Manuscript, Department of Mechanical Engineering, University of California at Berkeley (1993).
[13] J. Casey, "Geometrical derivation of Lagrange's equations for a system of particles," American Journal of Physics, 62, pp. 836-847 (1994).
[14] J. Casey, "On the advantages of a geometrical viewpoint in the derivation of Lagrange's equations for a rigid continuum," Journal of Applied Mathematics and Physics (ZAMP), 46 (Special Issue) pp. S805-S847 (1995).
[15] G. Coriolis, Theorie Mathematique des effets du Jeu de Billard, 1
Carilian-Gouery, Paris (1835).
[16] C. A. Coulomb, "TMorie des machines simples en ayant egard au frottement et ala roideur des cordages" 2 Memoires de Mathematique et de Physique presentes a l'Academie Royale des Sciences par divers Savans, et lus dans ses assembles, 10, pp. 161-332 (1785).
[17] H. Crabtree, An Elementary Treatment of the Theory of Spinning Tops and Gyroscopes, Third Edition, Chelsea Publishing, New York (1967).
[18] R. Cushman, J. Hermans, and D. Kemppainen, "The rolling disk," in Nonlinear Dynamical Systems and Chaos, edited by H. W. Broer, S. A. van Gils, I. Hoveijn, and F. Takens, Progress in Nonlinear Differential Equations and their Applications, 19, pp. 21-60. Birkhauser, Basel (1996).
[19] G. Darboux, Le~ons sur la Theorie Generate des Surfaces et les Applications Geometriques du Calcul Infinitesimal, 3 Parts 1-4, GauthierVillars, Paris (1887-1896).
[20] F. Diacu and P. Holmes, Celestial Encounters: The Origins of Chaos and Stability, Princeton University Press, Princeton (1996).
1The title of this book translates to Mathematical Theory of Effects on the Game of Billiards.
2The title of this paper translates to "Theory of simple machines with consideration of the friction and rubbing of rigging."
3The title of this book translates to Lectures on the General Theory of Surfaces and Geometric Applications of Calculus.
References 195
[21] R. Dugas, A History of Mechanics, (translated from French by J. R. Maddox), Dover Publications, New York (1988).
[22] L. Euler, "Recherches sur le mouvement des corps celestes en general," 4 Memoires de l'Academie des Sciences de Berlin, 3, pp. 93-143 (1749). Reprinted in Leonhardi Euleri Opera Omnia, Series Secunda, 25, pp. 1-44, edited by M. Schiirer, Ziirich, Orell Fiissli (1960).
[23] L. Euler, "Decouverte d'un nouveau principe de mechanique," 5
Memoires de l'Academie des Sciences de Berlin, 6, pp. 185-217 (1752). Reprinted in Leonhardi Euleri Opera Omnia, Series Secunda, 5, pp. 81-108, edited by J. 0. Fleckenstein, Ziirich, Orell Fiissli (1957).
[24] L. Euler, "Nova methodus motum corporum rigidorum determinandi,"6 Nova Commentarii Academiae Scientiarum Petropolitanae, 20, pp. 208-238 (1776). Reprinted in Leonhardi Euleri Opera Omnia, Series Secunda, 9, pp. 99-125, edited by C. Blanc, Ziirich, Orell Fiissli (1968).
[25] M. Fecko, "Falling cat connections and the momentum map," Journal of Mathematical Physics, 36, pp. 6709-6719 (1995).
[26] J.-F. Frenet, "Sur quelques proprietes des courbes a double courbure,"7 Journal de Mathematiques pures et appliquees, 17, pp. 437-447 (1852).
[27] T. D. Gillespie, Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale (1992).
[28] W. Goldsmith, Impact: The Theory and Physical Behavior of Colliding Solids, Arnold, New York (1960).
[29] D. T. Greenwood, Principles of Dynamics, Second Edition, PrenticeHall, Englewood Cliffs, New Jersey (1988).
[30] M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, San Diego (1981).
[31] J. Hermans, "A symmetric sphere rolling on a surface," Nonlinearity, 8, pp. 493-515 (1995).
4 The title of this paper translates to "Researches on the motion of celestial bodies in general."
5The title of this paper translates to "Discovery of a new principle of mechanics." 6The title of this paper transltates to "A new method to determine the motion of
rigid bodies." 7 The title of this paper translates to "On several properties of curves of double
curvature." According to Spivak [59], curves of double curvature is an old term for space curves.
196 References
[32] J. Heyman, Coulomb's Memoir on Statics: An Essay in the History of Civil Engineering, Cambridge University Press, Cambridge (1972).
[33] R. C. Hibbeler, Engineering Mechanics, Eighth Edition, Prentice-Hall, Upple Saddle River, New Jersey (1997).
[34] T. R. Kane and M. P. Scher, "A dynamical explanation of the falling cat phenomenon," International Journal of Solids and Structures, 5, pp. 663-670 (1969).
[35] E. Kreyszig, Differential Geometry, Revised Edition, University of Toronto Press, Toronto (1964).
[36] D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York (1989).
[37] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Fourth Edition, Cambridge University Press, Cambridge (1927).
[38] H. H. Mabie and F. W. Ocvirk, Mechanisms and Dynamics of Machinery, Third Edition, John Wiley & Sons, New York (1978).
[39] J. L. Meriam and L. G. Kraige, Engineering Mechanics: Dynamics, Fourth Edition, John Wiley & Sons, New York (1997).
[40] F. C. Moon, Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers, John Wiley & Sons, New York (1992).
[41] F. R. Moulton, An Introduction to Celestial Mechanics, Second Edition, Macmillan, New York (1914).
[42] Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, translated from Russian by J. R. Barbour, American Mathematical Society. Providence, Rhode Island (1972).
[43] I. Newton, Philosophiae Naturalis Principia Mathematica. Originally published in London in 1687. English translation in 1729 by A. Motte, revised translation by F. Cajori published by University of California Press, Berkeley (1934).
[44] I. Newton, The Mathematical Papers of Isaac Newton, 5, edited by D. T. Whiteside, Cambridge University Press, Cambridge (1972).
[45] 0. M. O'Reilly, "On the dynamics of rolling disks and sliding disks," Nonlinear Dynamics, 10, pp. 287-305 (1996).
[46] P. Painleve, "Surles lois du frottement de glissement," 8 Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, 121, pp.
8 The title of this paper translates to "On the Jaws of sliding friction."
References 197
112-115 (1895), 140, 702-707 {1905), 141, 401-405 {1905), and 141, 546-552 {1905).
[47] B. Paul, Kinematics and Dynamics of Planar Machinery, PrenticeHall, Englewood Cliffs, New Jersey {1979).
[48] I. Peterson, Newton's Clock: Chaos in the Solar System, W. H. Freeman and Company, New York {1993).
[49] E. Rabinowicz, Friction and Wear of Materials, Second Edition, John Wiley & Sons, New York (1995).
[50] W. F. Riley and L. D. Sturges, Engineering Mechanics: Dynamics, Second Edition, John Wiley & Sons, New York {1996).
[51] E. J. Routh, The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies, Seventh Edition, Macmillan, London {1905).
[52] E. J. Routh, The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, Sixth Edition, Macmillan, London {1905).
[53] M. B. Rubin, "Physical restrictions on the impulse acting during threedimensional impact of two rigid bodies," ASME Journal of Applied Mechanics, 65, pp. 464-469 {1998).
[54] A. Ruina, "Constitutive relations for frictional slip," in Mechanics of Geomaterials: Rocks, Concrete, Soils, edited by Z. P. Bazant, pp. 169-199, John Wiley & Sons, New York {1985).
[55] J. A. Serret, "Sur quelques formules relatives ala theorie des courbes a double courbure," 9 Journal de Mathematiques pures et appliquees, 16, pp. 193-207 {1851).
[56] I. H. Shames, Engineering Mechanics- Statics and Dynamics, Fourth Edition, Prentice-Hall, Upper Saddle River, New Jersey {1997).
[57] A. Shapere and F. Wilczek, "Gauge kinematics of deformable bodies," American Journal of Physics, 57, pp. 514-518 {1989).
[58] M.D. Shuster, "A survey of attitude representations," The Journal of the Astronautical Sciences, 41, pp. 439-517 {1993).
[59] M. Spivak, A Comprehensive Introduction to Differential Geometry, 2, Second Edition, Publish or Perish, Berkeley {1979).
9 The title of this paper translates to "On several formulae relating to curves of double curvature." According to Spivak (59], curves of double curvature is an old term for space curves.
198 References
[60] D. E. Stewart, "Rigid-body dynamics with friction and impact," SIAM Review, 42, pp. 3-39 (2000).
[61] G. Strang, Linear Algebra and its Applications, Third Edition, Harcourt Brace Jovanovich Publications, San Diego (1988).
[62] S. H. Strogatz, Nonlinear Dynamics and Chaos, with Applications to Physics, Chemistry, Biology and Engineering, Addison-Wesley, Reading (1994).
[63] D. J. Struik, Lectures on Classical Differential Geometry, Second Edition, Dover Publications, New York (1988).
[64] J. L. Synge and B. A. Griffith, Principles of Mechanics, McGraw-Hill, New York (1942).
[65] C. Truesdell, Essays on the History of Mechanics, Springer-Verlag, New York (1968).
[6?] C. Truesdell and R. A. Toupin, The Classical Field Theories, in Handbuck der Physik, 3/1, edited by S. Fliigge, Springer-Verlag, Berlin (1960).
[67] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Fourth Edition, Dover Publications, New York (1944).
[68] D. V. Zenkov, A. M. Bloch, and J. E. Marsden, "The energymomentum method for the stability of non-holonomic systems," Dynamics and Stability of Systems, 13, pp. 123-165 (1998).
Index
ABS, 163 acceleration vector
a, 120 a, 2, 13, 32, 111 ap, 122
angular acceleration vector a, 110 angular momentum
H, 125, 128, 138, 167 HA, 126 He, 94, 98 H 0 , 74, 98, 125, 167 Hp, 93, 98
angular momentum theorem particle, 75, 167 system of particles, 99
angular velocity vector w, 110 Antman S. S., 47, 193 Appell P., 122 arc-length parameter s, 2, 20, 24,
26, 27, 30, 32, 58, 60 Arnol'd V. I., 77, 193
balance law angular momentum, 101, 135,
137, 167
impulse-momentum form, 73, 169
linear momentum, 6, 14, 33, 52, 73, 95, 97, 101, 135, 137, 167
Barrow-Green J., 184, 193 Beatty M. F., viii, 107, 109, 193 Bedford A., vii, 187, 191, 193 Beer F. P., vii, 187, 191, 193 Bellman R., 129, 193 billiards, 121, 157 binormal vector eb, 21, 23, 25, 27,
29, 31 Bloch A. M., 153, 198 Bottema 0., 116, 193 bowling balls, 157 Brach R. M., 79, 84, 194
Cartesian coordinates, 6, 58, 60, 74, 96, 112, 181
Casey J., viii, 107, 109, 112, 128, 130, 131, 140, 194
center of mass rigid body, 118 system, 165
200 Index
system of particles, 92, 96, 98 chain rule, 13, 183 Chaplygin S. A., 153 Chasles' theorem, 109 circle, 27 circular helix, 29 circular motion, 3, 27 coefficient
dynamic friction J-ld, 43, 45, 152, 153
restitution e, 78, 80-84, 175 static friction J-L 8 , 43, 45, 46,
152, 153 collision, 78, 175
elastic, 81 example, 84, 85, 175 plastic, 81
compression, 79 cone, 44, 50, 77 configuration
present, 107 reference, 107
conservation angular momentum, 75-77,98,
100, 169, 173, 174, 176 energy,36,63,67,68, 77,102,
146, 151, 153, 170, 173, 176, 185
linear momentum, 74, 81, 95, 98, 169, 173, 176
mass, 119, 130 constraint
kinematical, 15, 121, 151, 153 rolling, 121, 151, 153 sliding, 121, 151, 153
Coriolis G., 121, 194 corotational basis, 112 Coulomb C. A., 41, 194 Crabtree H., 170, 194 curvature "'' 21, 25, 27, 28, 31 Cushman R., 122, 194 cylindrical polar coordinates, 11,
14,27,29,51,58,60, 74, 76, 77, 96, 100, 183
Darboux G., 23, 194 Darboux vector wsF, 23 density
p, 119 Po, 119
Diacu F., 184, 194 Dugas R., 41, 195
Euler angles, 109 Euler L., 6, 109, 137, 195 Euler parameters, 109 Euler's first law, 6, 14, 33, 52, 95,
135, 137 Euler's second law, 135, 137 Euler's theorem, 109
Fecko M., 170, 195 fixed-axis rotation
angular acceleration vector BEz, 111, 138
angular velocity vector BEz, 111, 138
corotational basis, 113 example, 116, 122, 147, 153,
157, 171 rotation matrix, 111, 113, 138
Flavin J. N., viii force
bearing, 157 central, 76 conservative, 60, 63, 77, 102,
151 constant, 61 constraint, 14, 15 friction, 42, 45, 46, 50, 65,
153, 155 gravity, 7, 14, 15, 33, 50, 61,
65, 77,149,155,157,172, 185
nonconservative, 61, 63 normal, 14, 15, 34, 42, 45, 46,
50, 65, 95, 100 reaction, 42, 95, 143, 149, 172 resultant, 6, 136, 143, 149, 167,
172
spring, 48, 50, 62, 65, 77, 95, 100, 149, 185
tension, 14, 15, 102 toppling, 146
Fowler W., vii, 187, 191, 193 free-body diagram, 7, 15, 96, 136,
149, 172 Frenet J.-F., 20, 195 friction, 41, 65, 67, 151-153, 177 Fufaev N. A., 121, 153, 196 fundamental theorem of calculus,
2
Gillespie T. D., 146, 195 Goldsmith W., 79, 195 gradient of a function, 60 Greenwood D. T., 109, 195 Griffith B. A., viii, 109, 198 Gurtin M. E., 107, 109, 128, 130,
195
harmonic oscillator, 185 helicoid, 29 Hermans J., 122, 153, 194, 195 Heyman J., 41, 196 Hibbeler R. C., vii, 187, 191, 196 Holmes P., viii, 184, 194 HookeR., 48 Hooke's law, 48 Huygens C., 64
impact, 78-80, 175 impulse
angular, 168 linear, 73, 80, 82, 168
inertia matrix, 127 matrix of cylinder, 129 moments of, 128 principal axes, 129 products of, 128 tensor, 126
initial conditions, 8, 16, 35, 36, 76,98, 145,151,157,184-186
Index 201
inverse function theorem, 2
Jacobi's elliptic functions, 151, 185 Johnston E. R., vii, 187, 191, 193
Kane T. R., 170, 196 Kemppainen D., 122, 194 Kepler J., 77 Kepler's problem, 76 kinetic energy T, 59, 66, 83, 94,
140, 149, 167, 170 Koenig decomposition
rigid body, 140 system of particles, 94
Koenig J. S., 140 Korteweg D. J., 122 Kraige L. G., vii, 77, 128, 130,
187, 191, 196 Kreyszig E., 22, 29, 196
Lawden D. F., 151, 185, 196 Leibniz G. W., 64 linear momentum G, 6, 14, 73,
81, 93, 118, 166 linear spring, 48 Love A. E. H., 47, 48, 196
Mabie H. H., 116, 196 Marsden J. E., 153, 198 matrix
determinant, 75, 109, 182 inverse, 118 positive definite, 129 proper-orthogonal, 109 rotation, 109 skew-symmetric, 110 symmetric, 129 transpose, 110
mechanisms four-bar linkage, 116 slider crank, 116
Meriam J. L., vii, 77, 128, 130, 187, 191, 196
moment applied torque, 157
202 Index
reaction, 138, 149, 172 resultant, 136, 143, 149, 155,
167, 172 Moon F. C., 184, 196 motion, 108 Moulton F. R., 77, 196 moving curve, 47
Naghdi P. M., viii Neimark Ju. 1., 121, 153, 196 Newton 1., 6, 48, 64, 76, 79, 196 Newton's second law, 6, 14, 33,
52, 95, 135, 137 Newton's third law, 6, 98 normal vector n, 44, 51, 80, 121,
151, 153
O'Reilly 0. M., 122, 153, 196 Ocvirk F. W., 116, 196 orthonormal, 181
Painleve P., 177, 196 parallel-axis theorem, 130 path C, 1, 31 Paul B., 116, 197 Peterson 1., 184, 197 planar pendulum, 14, 69, 185 plane
horizontal, 42, 44, 50, 122 inclined, 153 moving, 47 osculating, 22 rectifying, 22 rough, 47
plane curve, 23, 33 Poincare H., 184 pool, 121, 157 position vector
X,119 :X, 119, 167 r, 1, 12, 31, 115 rp, 121, 151, 153 x, 108, 110, 115
potential energy U, 6Q-62 power
force, 57, 142 moment, 142
principal normal vector en, 21, 25, 27, 28, 31
product cross, 182 dot, 182
product rule, 183 projectile problem, 7, 74, 185
Rabinowicz E., 43, 197 radius of curvature p, 21 rectilinear motion, 4, 26, 45, 58,
145 relative position vector
n, 126 7r, 126
relative velocity vector Vre!, 44, 45, 53
restitution, 79 right-handed basis, 6, 21, 114, 181,
184 rigid body motion XR(X, t), 108 Riley W. F., vii, 77, 187, 191, 197 Rivlin R. S., viii rolling, 121, 151, 153, 157 Roth B., 116, 193 rotor, 157 rough curve, 65 Routh E. J., 79, 121, 153, 157, 197 Rubin M. B., 79, 197 Ruina A., 43, 177, 197
Scher M.P., 170, 196 Serret J. A., 20, 197 Serret-Frenet
formulae, 22, 31, 47, 66 triad, 19, 31, 35, 47, 58, 60,
66, 68 Shames I. H., vii, 187, 191, 197 Shapere A., 170, 197 Shuster M. D., 109, 197 sliding, 121, 125, 151, 153 slip velocity, 154, 157 space curve, 19, 26, 31, 45, 65
speed, 1 Spivak M., 22, 195, 197 static friction criterion, 45, 46, 53,
152, 156 Stewart D. E., 79, 177, 198 Strang G., 129, 198 Strogatz S. H., 177, 184, 198 Struik D. J., 22, 29, 198 Sturges L. D., vii, 77, 187, 191,
197 surface, 44, 80, 121, 151, 153 Synge J. L., viii, 109, 198
tangent vector et, 21, 24, 27, 28, 31
tangent vectors t1, t2, 44, 51, 80 The Four Steps, 6, 139 theory of rods, 47 torsion T, 22, 25, 27, 29, 31 total energy E, 63, 102, 151, 153,
173 Toupin R. A., 130, 198 Truesdell C., viii, 6, 130, 136, 137,
198
Index 203
vehicle dynamics, 43, 146, 176 velocity vector
v, 120 v, 1, 13, 32, 110 Vp, 122
Vierkandt A., 122
whirling tube, 186 Whittaker E. T., 109, 151, 184,
198 Wilczek F., 170, 197 work, 57, 142 work done by friction, 66, 153 work-energy theorem, 59, 64, 66,
102, 141, 142, 170
Zenkov D. V., 153, 198
