References

1. R. Abraham and J. E. Marsden, Foundations of Mechanics, Second Edition, The Benjamin/Cummings Publishing Company, Reading (1978).
   The Big Book among modern expositions of analytical mechanics. I had planned to study it thoroughly but now I see that I will probably not have the time.
2. S. A. Antman, Nonlinear Problems of Elasticity, Springer-Verlag, New York (1995).
   Thorough and rigorous and heavy going.
3. V. I. Arnold, Methods of Classical Mechanics, Springer Verlag New York, Inc., New York (1978).
   Rigorous but readable. Maybe even the best of its kind. A product of the once-great Soviet mathematics community.
4. U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Applied Mathematics, Philadelphia (1998).
   The standard (as of 1998) book on the numerical solution of ordinary differential equations constrained to lie in a surface.
5. R. S. Ball, Theory of Screws, Cambridge at the University Press, Cambridge (1900).
   Old books like this one are rich sources of details which can only be accumulated through a lifetime of loving devotion to their subject matter. As a rich source of ideas they have been mined by modern kinestheticians, sometimes without adequate attribution.
6. M. Born, Stabilität der elastischen Linie in Ebene und Raum, reprinted as Chapter I in his Ausgewahlte Abhandlungen, vol. 1, Vandenhoeck & Ruprecht, Göttingen (1963).
   Born's "Ph.D." thesis. It shows even at this early date, back in classical mechanics, that he was a very, very, very smart guy.
7. O. Bottema and B. Roth, Theoretical Kinematics, Dover Publications, Inc., New York (1990).
   This book is another reminder of the stupendous edifice which these old-timers have created, of the work which went into its creation and the intimidating amount of work which would be required of us to master any reasonable part of what they have already done.
8. L. Brand, Vectorial Mechanics, John Wiley & Sons, Inc., New York (1930).
   Excellent.
9. L. Brand, Vector and Tensor Analysis, John Wiley & Sons, Inc., New York (1947).
   Not only an excellent exposition of vector and tensor analysis but also an excellent introduction to differential geometry, analytical mechanics and certain hypercomplex numbers. Though out of print, this book above all deserves reprinting by Dover.
10. W. L. Burke, Applied Differential Geometry, Cambridge University Press, New York (1985).
    A good, modern but commonsensical, book.
11. J. M. Cook, An Application of Differential Geometry to SSC Magnet End Winding, FERMILAB-TM-1663 [SSCL-N-720], Fermi National Accelerator Laboratory, Batavia (April 1990).
    Based on Born's work plus some classical differential geometry, mostly obtained from Brand.
12. J. M. Cook, Strain Energy Minimization in SSC Magnet Winding, IEEE Transactions on Magnetics, Vol. 27, No. 2 (March 1991).
    Further development of the work described in the previous reference.
13. J. M. Cook, User's Guide for GSMP, A General System Modeling Program ANL/MHD-79-11, Argonne National Laboratory, Argonne, IL (October 1979).
    GSMP was the result of a heroic effort to apply principles of general system theory to the modeling of magnetohydrodymic systems. For years it was the mainstay of an engineering design team at Argonne but it never got a foothold out in the mainstream where now one can assemble systems by moving boxes and arrows around on a GUI.
14. R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, Inc., New York (1953).
    The translation of a standard compendium of what was at the time most of applied mathematics, written from the standpoint of what later became known as functional analysis though Courant never did like the axiomatic approach, even to von Neumann's Hilbert space! This approach came to dominate the field. As far as axiomatics were concerned, von Neumann could take 'em or leave 'em. Courant, to his dying day, decided that he didn't want to take 'em and wished that more modern American mathematicians would share his (dis)taste.
15. B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin (1989).
    I only referred to this book because of its title. It contains none of the numerical analysis necessary to actually implement the direct methods, in spite of its publication in the Applied Mathematical Sciences series, but it does contain an exposition of the purely mathematical foundations under these methods.
16. J. P. Den Hartog, Strength of Materials, Dover Publications, Inc., New York (1949).
    A clearly-written introduction for engineers, and the rest of us. An expository classic.
17. L. P. Eisenhart, An Introduction to Differential Geometry with Use of the Tensor Calculus,Princeton University Press, Princeton (1947).
    This is not a good book. It got good reviews at the time, maybe by people who read his 1909 Differential Geometry of Curves and Surfaces in their youth, which may have been a good book for all I know. Maybe it was their introduction to Riemannian geometry and they have regarded it with affection ever since so when they reviewed the current book they didn't go through it in detail, relying instead on their memories of the earlier work. Often when busy mathematicians are asked to review a text in their field of expertise they impatiently skim over the given proofs because they already know that stuff. On that level this book looks pretty good. Of course I have done a lot (indeed!) of skimming of the books reviewed here without even being (indeed) an expert, but I did read all of Eisenhart, carefully, twice, going through every detail of the proofs because of its reputation and because I wanted to learn tensor analysis. That was a mistake. Better read Chapter IX in Brand's Vector and Tensor Analysis, as I did later. It contains just about all of the tensor analysis that we need to know, for a while. The tensors are often written just as themselves, without a lot of indices. This is the modern coordinate-free style, an advance that is not just typographical. For even fewer indices go on to read Chapters IX and XVI in Mac Lane & Birkhoff to see how the pure mathematicians do it, but only if that is to your taste. One thing about Eisenhart's book is that a paperback version is being kept in print by its original publisher, maybe out of nostalgia.
18. L. Euler, Additamentum I, De Curvis elasticus, pages 231-297 in Methodus Inveniendi Lineas Curvas, edited by C. Carathéodory, Orell Füssli Turici, Berne (1952).
    Just one of the products of an immensely prolific, cosmic genius.
19. Giampaolo Ferrari, About the Deformation of an Inner Cable on the Hard Way Bending Plane, (MathCad), Technical Division, Fermilab, Batavia.
    Giampaolo was a summer student who, with Kerry Ewald, did an experimental determination of the two flexural rigidities of a typical superconducting cable, finding that their ratio was, as I remember (the report seems to have been lost), approximately equal to six.
20. R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Addison-Wesley Publishing Co., Reading (1964).
    The 20^th century has given us several monumental multi-volume surveys of mathematical physics, the works of Frank and von Mises, Courant and Hilbert, Sommerfeld, Pauli, Landau and Lifshitz, Morse and Feshbach, Bob Hermann, Feynman, ... Sometimes informal and sometimes, well, monumental, each bore the imprint of the personality of its author. Feynman's is both informal, as based on his lectures, and monumental, as an edited extensive (enlarged and refined by his co-authors?) survey. Though the course was not a success (?why?) these volumes are treasured favorites of the English-speaking (and English-reading, which is practically everybody) physics community. Feynman was a physically intuitive genius, far ahead of the mathematicians who are still trying to understand, in their own way, his renormalization theory and integration on function space.
21. G. R. Fowles, Analytical Mechanics, Fourth Edition, Saunders College Publishing, New York (1986).
    A conventional but good introduction to the subject, and it is still in print, unlike many of my other references here.
22. P. Franklin, Methods of Advanced Calculus, McGraw-Hill Book Company, Inc., New York (1944).
    A standard but very good advanced calculus text, undoubtedly permanently out of print.
23. I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Inc., Englewood Cliffs (1963).
    Gelfand is one of the top mathematicians of the Twentieth Century yet this exposition of well-selected, rather advanced parts of the Calculus of Variations is quite readable. It continues the Soviet tradition of clear expositions of deep mathematics. And in the original Russian their books were cheap too. Subsidized by the state for propaganda purposes? Yes, but so what! Us poverty-stricken American students benefitted. Now, alas, the Soviet Union no longer exists. Too bad.
24. The Holy Bible, Revised Standard Edition, Thomas Nelson & Sons, New York (1953).
25. W. C. Graustein, Introduction to Higher Geometry, The Macmillan Company, New York (1946).
    Uninspired, but this was my introduction to one of the most beautiful subjects in all of mathematics. I studied it lovingly and it is still my favorite projective geometry text and reference. (The Nobel Prize winning physicist Julian Schwinger did not like it at all.)
26. H. W. Guggenheimer, Differential Geometry, Dover Publications, Inc., New York (1977).
    Another of Dover's good introductions to the subject.
27. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, Inc. New York (1978).
    Has a reputation of being one of the nicest introductions to the fields mentioned in its title.
28. R. C. Hibbeler, Mechanics of Materials, Second Edition, Macmillan College Publishing Company, New York (1994).
    A very good textbook but there must be many others. Pick your own.
29. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Revised Edition, American Mathematical Society, Providence (1957).
    A big book so it contains a fairly complete introduction to functional analysis, which may be expositorily somewhat out of date. It is kept in print by the American Mathematical Association and is even downloadable at http://www.ams.org/online_bks/coll31/ but you could probably do better with something more recent.
30. T. von Karman and M. A. Biot, Mathematical Methods in Engineering, McGraw-Hill, New York (1940).
    Von Karman was a powerful and versatile applied mathematician and it shows even at this expository level.
31. E. Kreyszig, Differential Geometry, Dover Publications, Inc., New York (1991).
    One of several good, standard (i.e., not ultra-modern), now-inexpensive introductions.
32. C. Lanczos, The Variational Principles of Mechanics, Fourth Edition, University of Toronto Press, Toronto (1970); reprinted by Dover Publications, Inc., New York (1986).
    A thoughtful book which goes beyond the formalism. For example the derivation of Euler's equations for rigid body motion from d'Alembert's principle may make you think. Actually I do rather wish that for greater clarity he had acquired some of the modern pure mathematician's desire to explicitly formalize all of the concepts used but I suppose that that would have directed attention away from his deeper aim, and in analytical mechanics it may even be impossible. Has the concept of force ever been completely formalised? Dover did well, as usual, to select this historically important exposition for reprinting.
33. L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Second English Edition, Pergamon Press, New York (1970).
    Landau is one of the top theoretical physicists of the Twentieth Century. This book is Volume 7 of an in-depth multi-volume survey of all of theoretical physics, yet the mathematical derivations are intuitive and (considering their aims) as easy to follow as one could ever hope for.
34. L. D. Landau and E. M. Lifshitz, Mechanics, Third Edition, Butterworth-Heinemann, Oxford (1976).
    This slim book is slim because it gets right to the conceptual heart of the subject. For example it contains no ε-δ proofs or slick manifolds. Nor, less fortunately, does it contain any chaos theory or nonlinearity, even to the extent that these theories were known by the time of Poincare.
35. T. Levi-Civita, The Absolute Differential Calculus, Dover Publications, Inc., New York (1977).
    A classical text written by the man who invented parallel displacement.
36. M. M. Lipschutz, Theory and Problems of Differential Geometry, Schaum's Outline Series, McGraw-Hill, New York (1969).
    Standard contents for an introduction. A typically good Schaum's Outline.
37. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Fourth Edition, Dover Publications, New York (1927).
    The classical exposition of the theory of elasticity, product of a lifetime of work on the subject, produced at a pinnacle in the development of the theory, all at the Dover paperback price.
38. Saunders Mac Lane and Garrett Birkhoff, Algebra, Third Edition, Chelsea Publishing Company, New York (1993).
    This is the 20^th century's final abstract algebra text, the wrap-up of a series of texts which began with van der Waerden's seminal "Moderne Algebra", followed in America by Birkhoff and Mac Lane's famous "A Survey of Modern Algebra" which spawned in turn a host of follow-ons. Now that surge has abated (has the subject been exhausted?) and has been finalized by the same two authors who started it. Mac Lane, the grand old man of American abstract algebra, is the co-founder and chief advancer of category theory (mathematical, not philosophical category theory) , which abstracts from all of pure mathematics and can even be taken as a foundation of computer science although this conquest is being resisted. The order of the authors names was reversed from that of the "Survey" because of Garrett's many other deep interests and contributions to both pure and applied mathematics.
39. J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs (1983).
    High-brow. Highly mathematical. Modern. Dependable.
40. J. C. Maxwell, The Scientific papers of James Clerk Maxwell, vol. I, Dover Publications, Inc., New York (1965).
    Another great genius, of course.
41. A. Mishchenko and A. Fomenko, A Course of Differential Geometry and Topology, Mir Publishers, Moscow (1988).
    Why did I include this book when it is no longer obtainable in this country? Well, for one thing I own it myself and this whole Web document was written mostly for my own benefit. For another as a memento to the great Soviet tradition of advanced but clear and inexpensive mathematical expositions. (Rumor has it that the authors were paid by the word, which discouraged terseness.) And this book follows differential geometry out of the Nineteenth Century into the Twentieth where the older stuff was cleaned up and good new work was done. There is a companion book Problems in Differential Geometry by Mishchenko, Solovyev and Fomenko which I have not seen; and a more advanced, three volume, work Modern Geometry by Dubrovin, Fomenko and Novikov, which I have not seen either but which is available in the United States. Its Second Edition was published in 1993 by Springer.
42. A. V. Pogorelov, Differential Geometry, P. Noordhoff N.V., Groningen (No printing date given in the book! Isn't that illegal?).
    An introductory text by a very advanced Soviet geometer. You probably won't be able to find it anywhere anymore, the product of a great but fading tradition. I got it as a discard from the Argonne mathematics library, also fading as an archival repository of great old mathematics books.
43. H. Pottmann and J. Wallner, Computational Line Geometry, Springer (2001).
    As promised by the title, this book is a good exposition of line geometry and hence of the ruled surfaces that we need, though at such a high algebraic level that their word "computational" does not get us all the way down into the numerical analysis we need to write code. The authors' real numbers give them infinite precision.
44. W. H. Press, B. P. Flannerey, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in FORTRAN, Second Edition, Cambridge University Press, New York (1992).
    The authors were not specialists in the field of numerical analysis so they took their lumps in reviews by the reigning authorities, but they were experts in the uses of numerical analysis and so they related the mathematics to its applications. The authorities had to pay attention, envious attention, to the book when they noticed it beginning to appear on the desks of engineers. By now it has become the de facto standard in its big niche.
45. P. J. Rabier and W. C. Rheiboldt, Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint, Society for Industrial and Applied Mathematics, Philadelphia (2000).
    A good modern treatment. Though its subject is rigid body motion, by Kirchhoff's kinetic analog^xx the mathematics translates perfectly over to our elasticæ. I depended heavily on this book while trying to constrain a 3-dimensional elastica onto the surface of a cylinder. The acronym "DAE" means "differential-algebraic equation", see Ascher and Petzold above.
46. H. I. Rosten, The Constant Perimeter End, RL-73-096,Applied Physics Division, Rutherford Laboratory, Chilton Berks (September, 1973).
    The paper with which we started.
47. A. Schopenhauer, The World as Will and Representation, Dover Publications, Inc., New York (1966).
    An unexpectedly readable book by a personally nasty, post-Kantian, anti-Hegelian pre-Nietzscheite.
48. J. M. Selig, Geometrical Methods in Robotics, Springer-Verlag New York, Inc., New York (1996).
    A modern but not hypermodern book on the kinematics and dynamics of linked rigid bodies. (These links correspond to the Δs's of our difference approximation to the differential equation determining the base curve.) I have not studied this book systematically at all but have looked around in it quite a bit. From what I have seen it is very good. We live in an era of superb textbooks. Not only are the classics of the past available but present-day mathematicians have been busy organizing and unifying that past into a (slick?) modern form.
49. S. Sternberg, Lectures on Differential Geometry, Second Edition, Chelsea Publishing Company, New York (1983).
    A modern treatment by a heavyweight in the field.
50. D. Struik, Lectures on Classical Differential Geometry, Second Edition, Dover Publications, Inc., New York (1950).
    As the title says, "Classical". (Note: The birthdate "1894" on page ii of the book must now be completed to "1894-2000"!)
51. S. P. Timoshenko, History of the Strength of Materials, Dover Publications, Inc., New York (1953).
    This famous (Russo-)American elasticity theorist is also a scholar. Here he puts the theory into a human context, showing the struggles with which concepts were created that we take for granted because they were handed to us fully formed.
52. V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations, Dover Publications, Inc., New York (1959).
    Courant and Hilbert didn't give Volterra proper (actually: any) credit.
53. C. E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge University Press, Cambridge (1955).
    Old but good, and out of print, probably permanently out of print.
54. E. P. Wigner, The unreasonable effectiveness of mathematics in the physical sciences, Communications in Pure and Applied Mathematics, vol 13, 1960.
    The much-referred-to article in which this Nobel Prize winner mulls over "The enormous usefulness of mathematics in the natural sciences ... something bordering on the mysterious ... it is difficult to avoid the impression that a miracle confronts us here."