Book Description

Customer Reviews (5)

Master Piece
As the other reviewers have said, this is a master piece for various reasons. Lanczos is famous for his work on linear operators (and efficient algorithms to find a subset of eigenvalues). Moreover,he has an "atomistic" (his words)view of differential equations, very close to the founding father's one (Euler, Lagrange,...).

A modern book on linear operators begins with the abstract concept of function space as a vector space, of scalar product asintegrals,... The approach is powerful but somehow we loose our good intuition about differential operators.

Lanczos begins with the simplest of differential equations and use a discretization scheme (very natural to anybody who has used a computer to solve differential equations) to show how a differential equation transforms into a system a linear algebraic equation. It is then obvious that this system is undetermined and has to be supplemented by enough boundary condition to be solvable. From here, during the third chapters, Lanczos develops the concept of linear systems and general (n x m) matrices, the case of over and under determination, the compatibility conditions, ...
It is only after these discussions that he returns (chapter 4) to the function space and develops the operator approach and the role of boundary conditions in over and under-determination of solutions and the place of the adjoint operators. The remaining of the book develops these concepts : chp5 is devoted to Green's function and hermitian problems, chap7 to Sturm-Liouville,... The last chapter is devoted to numerical techniques, amazing if one think that the book was written at the verybeginning of computers, which is a gem by itself.

Lanczos again
Somebody writen:
"Some mathematics and physics writers stand head and shoulders above the rest. Goldstein...Liboff...Morrison...Morse and Feshbach...and Lanczos. A joy to read, if you are both mathematically and verbally inclined."

I think some mathematics and physics writers stand head and shoulders above even Goldstein...Liboff...Morrison...Morse and Feshbach. It is the case of Lanczos and Dirac.

wonderful book, elegantly written
This book has material I've found in no other book. Lanczos is a pleasure to read -- his writing is clear, elegant, and entertainingly opinionated. I've liked every book of his that I've read.

A must!
A very intuitive (geometrical) exposition of matrix calculus, adjoint problems, bilinear identity and Green's function (and more). If you really want to understand these concepts, read this masterpiece!

A joy to read.
Some mathematics and physics writers stand head and shoulders above the rest.Goldstein...Liboff...Morrison...Morse and Feshbach...and Lanczos.A joy to read, if you are both mathematically and verbally inclined. ... Read more

Book Description

Customer Reviews (12)

a lot of unfamiliar variational tricks, sometimes lacks proofs or underexplains
I've read this gem and done most of the evercises in about 3 months. Before that legendary book I'd had the usual crappy course in Classical Mechanics based on Goldstein. The bottom line is the book will show you a lot of advanced material and unfamiliar manipulations. On the other hand there are sometimes statements lacking proof or more detailed lucid explanation. The book is appropriate for readers that already know what action is, totall beginners will be too shocked by the new concepts and won't be able to pick up the important nuances revealed by Lanczos.

Lanczos work clarified some of the concepts in which my CM course failed:
- the important difference in treating holonomic and nonholonomic constraints
- exact constraints are mathematical idealization of infinitely rigid constraint forces
- Lagrange multipliers for functionals (actions) not only functions
- the logical thread virtual work -> d'Alembert -> Hamilton's principle
- the connection between the action in configuration space andin phase space

The book introduced me to topics not covered by the course, which was my initial goal:
- elimination of ignorable variables in L or H formulation
- canonical transformations, definition and importance
- generating function of canonical transformation
- test for canonicity of transformation using Poisson brackets
- integral invariants of canonical transformations
- Hamilton's principal function
- Hamilton-Jackobi equation and analogy with optical wave surfaces
- separation of variables in H-J equation
- action-angle variables for separable periodic systems
- evolution of the system as a sequence of canonical transformation
- introducing geometry and geodesics in phase space

The reading definitely increased my freedom in manipulating the variational problem into equivalent variational problem. Examples of the two most weird for me manipulations are in the appendices. In the first appendix the Hamiltonian formulation is derived from the Lagrangian by introducing new variables, constraints and corresponding Lagrange multipliers, and then eliminating the variables. In appendix II, the most popular cases of Noether's theorem are derived by introducing new field variables in the action - I had no idea that was allowed. Very interesting was the idea that the world line of the system in configuration space can be parametrized with arbitrary parameter and the time becomes a function of that parameter that is varied together with the other generalized coordinates. Such variation is normal for GR but I've never seen it done in non-relativistic mechanics.

Some of the other reviews described the book as 'lucid'. I find that eggagerated - although the book shows lots of unfamiliar manipulations, sometimes proofs of validity or the necessary more detailed conceptual or calculational explanations are lacking. An example is the inclusion, all of a sudden, of the time as variable to be varied - where is the proof one is allowed to do that? In another case, the book tells you that by nullifying the boundary term when varying the action, one gets 'natural' boundary conditions for the Euler-Lagrange diff. equations. I failed to see how the physics of the problem would demand exactly those boundary conditions. Where the analogy between mechanics and optics was discussed, the book creates the impression it derived the Fermat's principle but in reality it simply proved that the path following the gradient of of constant surfaces is shortest between two points. So there is a certain gegree of fuzziness on calculational level (lacking proofs of validity) or conceptual level (underexplained concepts and relations).

I liked the the abundance of historical notes. You will learn that there are several formulations of the least action principle - Euler and Lagrange version, Jackobi version and Hamilton version. Each subsection has a small summary and there are a few problems per section to illustrate the main ideas but not enough for exercises.

There are two chapters that I think appeared in later editions and are too sketchy compared to the book core:

Chapter 9 discusses special relativity where you can see that guessing the relativistic Lagrangian on general grounds of Lorentz invariance gives almost effortlessly the relativistic dynamics without the usual gedanken experiments. At the end, Lanczos dives a little into GR using the Schwartzchild metric to derive orbits, bending of light rays and gravitational redshift around spherical body.

Chapter 11 gives a short presentation of fluid mechanics (a little unclear derivation, in Lagrange and Euler coordinates), elasticity, and electromagnetism. Noether's principle is used to derive the canonical and the symmetric energy momentum tensor. I haven't seen a crystal clear derivation of Noether anywhere and Lancsoz is not an exception. The problem is as usual ommiting what exactly is being transformed and why is that allowed.

Timeless classic, masterful ...
If you ask 10 PhD scientists: "Why is Schrodinger's Equation complex?" (contains the square-root of minus one), 9 out of 10 won't be able to give you the correct answer.

It has little to do with taking the root of negative numbers. After reading Lanczos you will know it has do with "space" and what is a proper physical law. (Now you have to read the book to parse this sentence. Good.)

This is one of many wonderful insights Lanczos provides; with humor, wonder and crystal clarity. This is not a 'text book' on mechanics, you will get more out of it if you are familiar with the subject. He gives you understanding, not technique.

It is as if you can hum a few tunes. Reading Lanczos is experiencing the entire opera for the first time. Now you know the full story, how each aria is a part of the fabric; how each fits in the situation, the motivation behind it. The tunes you liked become richer, more profound, they are connected. The next time you sing you fancy you are a Caruso, a Puccini.

It is so rare to encounter a master who is also a gifted writer.

Some reviewers compare Lanczos to Feynman's Lectures, I agree partly.Lanczos is more literate and much more humble. Feynman is so busy being the genius from Brooklyn that his exposition is choppy and uneven.Lanczos is a better organizer and writer.

Delightful ... simply brilliant
From organization, to prose, to content, to price, this is the best book on the Hamiltonian and Lagrangian formulation of Classical Mechanics.I just wish this book treated more subjects!The numbered list organization with pithy summaries really works for me.The thought provoking and mathematically fluent prose style is a joy to experience.The author is clearly a master of Einsteinian Relativity, Classical Physics, Differential Geometry, and function analysis.In fact I seem to recall him writing some other books along those lines.Lanczos is a real treat to read.I have read parts of over a dozen different books on Intermediate/Advanced Classical Mechanics and the things the Lanczos covers are just supperb.As a standalone text, it may not be the best choice, but when accompanied by Arya or Hand and Finch it is very enriching.FLuent and cohesive are the words that come to mind when describing this work.This book is especially good for someone who knows a good deal of math and would like to be introduced to classical mathematical physics.

I heartily recommend Lanczos's masterpiece!

So beatiful that feels like art
Lanczos makes mechanics feels like art in this superb work. Analytical Mechanics is the foundation of physics and Lanczos has complete command of the theme. The purpose of this book is to make one understand mechanics "from inside" and not to stress methods of problem solving. Lanczos says that very clearly in the preface. The beauty of the book is that it's not in the same category as Goldstein, instead feelink more likely to Landau, so the bad criticism of the 2-star guy comes from someone that missed this.

OK, but old-fashioned, few examples, and not many diagrams
.
This was probably a good book in its day (1950-1970), but
it's really old-fashioned now.A lot has happened in the
field of mechanics since Lanczos wrote it.For example:

- Computers are now used extensively to analyze and
simulate mechanical systems.

- The modern language of mechanics is much more geometric
and independent of any particular choice of coordinates.
If readers stop at Lanczos, they will have trouble
understanding the modern literature.He doesn't even
distinguish between vectors and one forms.

- Dynamical systems theory / qualitative dynamics has
contributed a lot to the understanding of mechanics
in the past 30 years.You won't read anything about
stable/unstable manifolds or strange attractors in
Lanczos.

The "problems" are so easy that they border on the
ridiculous.And don't try finding them at the end
of each chapter --- this book predates modern textbook
format.Lanczos hides his problems like Easter eggs.

In conclusion, this book is of historical interest only.
If you want to learn about modern mechanics, read
something that was published recently.

(I should add that the book is well-written, but that
doesn't fix the fact that it is dated.) ... Read more

Customer Reviews (5)

Master of Exposition
It's an excellent book.The best parts for
we were the chapters on Matrices and on
Harmonic Analysis. An outstanding aspect
of the latter chapter is Lanczos's exposition
of the motivation behind the Fourier integral
(transform) and its basic theory.The quality
of the writing is superb, very classical
and lucid.

It cannot, of course, serve as a textbook.
But if you're taking a Fourier theory
course using Stein and Shakarchi's book, say,
as I am currently, then it's a very handy
book that can complement abstract theory
with physical intuition.

very fine but could be more advanced
Lanczos' work is a fine, thorough text that covers most areas of advanced analysis in a readable style. His derivations are clear, his tangential explorations are absorbing, and he cites practical examples. The one area in which I find the book weak is harmonic functions, potential theory, and the applications of these to the calculus of resides. Consequently, the book is not "one-shop stopping" for all the mathematical techniques that an electrical engineer or physicist might require in his bag of tricks....

If you don't want just recipes...
Then this is the best book. Well, Hamming's is also so good! For Fourier analysis, and the taming of the Gibbs phenomenon, go straight to Lanczos. He knew it all, and was one of the inventorsof the fast Fourier transform. This book is in the class of Sommerfeld's "Partial Differential Equations of Physics" and Lighthill's "Fourier Analysis and Generalizaed Functions". This is a very high compliment. Did you know hewas also a first rate physicist, and a pioneer of quantum mechanics?

Simply the best book on numerical analysis
My dissertation advisor introduced me to this book over thirty years ago.I have since read it in its entirety twice and it is still the first book I consult when confronted with a new mathematical problem.

Lanczos'sunderstanding of applied mathematics is very deep and he has a rare way ofexplaining things clearly yet concisely.I find his description of linearsystems in terms of multidimensional coordinate systems, both orthogonaland skewed, to be the best anywhere.Also, his understanding andexplanation of harmonic analysis (he invented the FFT after all) is worththe price of the book by itself.

Buy it, read it (at least once) then seeif really need any other book on applied mathematics.

A Gem of an Applied Math Book
While this booked is dated because it was written for the days ofmechanical calculators, it contains a great deal of very useful material.His discussion of Chebyeshev Polynomials one of the best I seen. Hisdiscussion on telescoping of power series is one of the few available. Hegives great insight into a host of numerical methods. A very valuable workfor the computer age as well. ... Read more