The Variational Principles of Mechanics (Dover Books on Physics and Chemistry)

Book Description

Philosophic, less formalistic approach to perennially important field of analytical mechanics. Model of clear, scholarly exposition at graduate level with coverage of basic concepts, calculus of variations, principle of virtual work, equations of motion, relativistic mechanics, much more. First inexpensive paperbound edition. Index. Bibliography.

Customer Reviews:

a lot of unfamiliar variational tricks, sometimes lacks proofs or underexplains.......2007-07-18

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 and in 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 ..........2006-12-20

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.......2006-12-01

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.......2006-12-01

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.......2003-04-30

.
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.

Book Description

Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques, with large number of problems, from routine manipulative exercises to technically difficult assignments.

Customer Reviews:

One of the best books I've studied on invariant variational principles........2007-07-29

I bought this book six years ago for the exercises on tensor analysis and differential forms and it has become one of my favorite texts on invariant variational principles. The authors develop their account of tensors in a clear and logical manner, with several diagrams in the first few chapters to highlight geometrical concepts. The authors provide a commentary in words to accompany the mathematical exposition. I appreciate their pointing out the various implications of the results they develop and I found the exercises helpful and a useful aid to the development of the theory.

Chapter 6 on the calculus of variations for invariance problems under coordinate transformations is one of the best I have studied. Their explanation of the theorems of Noether and reference to Caratheodory's work clearly explain some of the deeper concepts of invariance. The authors include worked examples which demonstrate the application of the ideas as they are developed. Variational principles are developed further in subsequent chapters. Chapter 8 describes the application variational principles to invariant field theories and the Einstein vacuum field equations.

Cryptic.......2007-02-23

I think I have a decent handle on the prerequisites for this subject, and I frankly found this to be cryptic. Perhaps the style is a bit archaic; but, I found some of the general relativity texts to give a better introduction to the sujbect of tensors. I would not recommend this as an introductory text.

Helpful for the advanced student.......2007-01-30

This is a helpful book for the advanced undergraduate/graduate student. It contains difficult mathematics, so it is a good overview of the matter, but not for the enthusiastic "layman" (pardon my language). This is because a lot of math knowledge is presupposed.

Not for self study.......2006-04-21

I bought this book togheter with Kay - Tensor analysis - when I began to study tensors. Kay could be not a masterpiece, but you can use it to learn the subject with some efforts but without a teacher. Don't try to do the same thing with Lovelock. May be this book is very profound, but you have better avoiding it as a self study text.

A MUST, if one wants to learn the subject.......2005-09-11

The two authors must be experts in this field. Without wasting words, they always well motivate, why something is done. It starts at a gradual pace, and is never extremely demanding mathematically.
It is rare, that one finds a flawless book, but this is certainly one!

Average customer rating:

A useful reference text for gauge theory.
Spivak on Steroids
A mathematically tight and well-motivated work
Clean and modern exposition
This book is all that is bad about abstract mathematical physics writing

Gauge Theory and Variational Principles
David Bleecker
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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Waves & Wave Mechanics | Physics | Science | Subjects | Books

ASIN: 0486445461

Book Description

Detailed and self-contained, this text supplements its rigor with intuitive ideas and is geared toward beginning graduate students and advanced undergraduates. Topics include principal fiber bundles and connections; curvature; particle fields, Lagrangians, and gauge invariance; inhomogeneous field equations; free Dirac electron fields; calculus on frame bundle; and unification of gauge fields and gravitation. 1981 edition

Customer Reviews:

A useful reference text for gauge theory........2007-07-28

A useful reference text for gauge theory. While Bleecker includes an introductory chapter to cover prerequisites (tensor analysis, differential forms, Lie groups etc) the text assumes some familiarity with these techniques and gauge theory. The text is in the "definition, theorem, proof" format. Bleecker gives fairly detailed proofs which help the reader to follow most steps. The text is short and the price reasonable, so the absence of second quantization gauge potentials, for example, is perhaps understandable. While there are chapters on gauge invariance and Action density I found the material didn't go into the detail I was looking for on invariance problems (discussion of theorems of Noether, Caratheodory's methods, Lie algebras and groups). I rated this text lower because of the complete absence of exercises and the limited references and bibliography (about three pages in total).

Spivak on Steroids.......2007-02-24

As the title suggests this "text" serves as an introduction to the QFT and guage theories recast in the "modern" mathematical setting of differential geometry.
This book is only 167 ( 1/2 regular size paper ) pages long. Although self-contained I highly recommend the reader have a working knowledge of QFT and at least an introductory course in GR. The mathematical tools of the reader should include a course in analysis on manifolds at the Spivak level or higher, acquintance with fibre bundles and basic lie groups. For example in the first chapter ( 22 pages ) the author covers differential forms, manifolds, Stokes theorem, lie derivative, deRham cohomology, lie groups and algebras. The next 20 page chapter covers principal bundles and connections ( 3 definitions all shown to be equivalent and these turn out to be the physical equivalent of gauge potentials ) followed by lie algebra valued forms, exterior covariant derivative curvature and the structure equation.
Chapter 3 defines particle fields as mappings from the principal bundle to a vector space which are equivariant or to the space of sections of the associated vector bundle. We now see that guage transformations are nothing more but the automorphisms of the bundle with certain requirements. Langrangians are developed as mappings of the space of 1 jets to the reals. G invariance of langrangians is defined and is shown to be an insufficient criteria for invariance under gauge transformations. However, we see that introduction of a connection and hence covariant exterior derivative on the bundle our Langrangian becomes gauge invariant.
The next chapter introduces action densities and shows that a particular particle field obeys the principle of least action iff it is stationary which is true iff Langrange's equation holds. There is alot of mathematical notation and machinery developed here. At this stage spin zero electrodynamics are treated.
Current are defined on PFB and a conservation of charge for g-invariant Langrangians with stationary particle fields is shown to be true. This chapter introduces the "self interaction" term for the gauge field and shows that a particle field and connection ( gauge potential ) obey the principle of least action iff they satisfy BOTH langrange equation and the inhomgeneous field equation.
The next chapter introduces spinor bundles ( to add spin particles to our repretoire ) requiring modifications of the previous mathematical tools leading to the appropriate Langrangian. Lagrange's equation is shown to reduce to the dirac free field equation.
The next chapter shows how to deal with interactions between the particle fields with spin and guage fields. This requires "spliced bundles" ( one where the particles with spin live and the other where guage fields live ) which requires another straight forward modification of our mathematical tools...redefining particle fields, Langrangians, currents, and action densities in the process. We see that in this general setting the particle fields and gauge potentials ( connection ) are stationary ( satisfy principle of least action ) iff they satisfy two generalized versions of lagranges equation and an inhomegeneous field equation. The author shows how these reduce the the special cases of the dirac electron field and yang mills field to the dirac and yang-mills equations and the inhomegeneous maxwell and yang-mills equations, respectively.
Chapter 8 goes over the mathematics of general relativity in about 10 pages.
Chapter 9 attempts to unify gauge theories and gravitation showing that the Einstein field equations and the Yang-Mills equations follow from a single variational principle dependent upon the scalar curvature of the metric defined on a suitable PFB. Problems of this unification are explained.
The final chapter explores symmetry breaking monopoles and instantons.

As you can see there is alot to absorb with the prerequisites noted above.
The book has no examples or exercise but each theorem is proved. There is also a page with corrections which is refreshing and a page summarizing the notation and page they are first introduced.

A mathematically tight and well-motivated work.......2006-07-06

Professor Bleecker has succeeded in writing a book for mathematicians and physicists. And, it's all there. I would rate this work 5-star, except I fear some physicists might find the mathematical format a bit tough (definition, theorem, lemma, etc.) As a mathematician studying physics I hope I am wrong. I find this book user-friendly due to its formality and "compactness". I caution those w/o a fair degree of mathematical acumen that this big, little book is a good deal more formal than, say, Gilmore's "Lie Groups, Lie Algebras, and Some of their Applications." But, then we all must bite the bullet. With effort, I think you will find this a chewable bullet.

Clean and modern exposition.......2006-06-19

Despite what another reviewer said, this book uses standard differential geometry notation. The notation is of the invariant (no index) style of Kobayashi and Nomizu.
I find it a delightful little book. It should be good for anyone with a background in manifold theory and Lie groups.

This book is all that is bad about abstract mathematical physics writing.......2006-05-16

Am I new to gauge theory: no.
Why does he make me feel I am?

Are these new generalized symbols really necessary to his treatment.
Apparently the author believes that rather than use words to explain
theorems...
He can rely almost entirely on this new set of symbols
that he has used to translate gauge theory to fiber bundles.
Why is it that so many times
the reader is the one who is made to do the work
of teaching himself and learning new "languages"
when the author claims he will be teaching you?

I feel that I have to warn the reader, that although he claims to be "mainstream"
differential geometry, he lies.
he uses very non-standard ( at least for the papers I've read)
notation and instead of starting at the simple gives the most abstract examples
of the notation even in chapter "0".

I grade this book as a "F"... way below even the worst
group theory book I have by an English set of professors!
I'm really as sorry as I can be, but he got me for the money,
by false advertising...
It is a very badly written book,
more a "show off"... see what I can do book,
than a I will teach you to understand book.
I had to go back to other books to compare the notation
so many times that I stack them with this book.
I spent half a year plotting solitons in Mathematica,
and I don't recognize anything in his treatment of them.
Monopoles and Instantons are even worse in this book, if possible.
It is not that I I don't believe he is using correct mathematics,
it is just that it is so hard to tell if he is!
Roger L. Bagula

Variational Principles in Dynamics and Quantum Theory (Dover Books on Physics)

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Variational Principles in Dynamics and Quantum Theory (Dover Books on Physics)
Wolfgang Yourgrau , and Stanley Mandelstam
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486458881

Book Description

Focusing on applications relevant to modern physics, this text for advanced undergraduates and graduate students surveys variational principles, examining their relationship to dynamics and quantum theory. It stresses the history and theory of these concepts rather than their mechanics, providing many insights into the development of quantum mechanics. 1968 edition.

Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators: Including Seminal Papers of Julian Schwinger (Particle Acceleration and Detection)

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Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators: Including Seminal Papers of Julian Schwinger (Particle Acceleration and Detection)
Kimball A. Milton , and Julian Schwinger
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540292233

Book Description

The present volume comprehensively summarizes the late Julian Schwinger's work on the theory of electromagnetic radiation and its application to waveguides, transmission lines, accelerator physics and synchrotron radiation. The first part, written in textbook style, has grown out of lectures and manuscripts by Julian Schwinger prepared during the war at MIT's Radiation Laboratory, updated with material developed by Schwinger at UCLA in the 1970s and 1980s, and by Milton at the University of Oklahoma since 1994. The second part consists of reprints of more than 15 papers (some of them never published or available with very limited circulation) by Schwinger on these topics. This volume will be of great interest to physicists, electrical engineers and applied mathematicians whether they be students or seasoned professionals. A paperback study edition of the first part is available separately.

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Variational Principles in Physics
Jean-Louis Basdevant
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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A Geometric Approach to Differential Forms

ASIN: 0387377476

Book Description

Optimization under constraints is an essential part of everyday life. Indeed, we routinely solve problems by striking a balance between contradictory interests, individual desires and material contingencies. This notion of equilibrium was dear to thinkers of the enlightenment, as illustrated by Montesquieu’s famous formulation: "In all magistracies, the greatness of the power must be compensated by the brevity of the duration."

Astonishingly, natural laws are guided by a similar principle. Variational principles have proven to be surprisingly fertile. For example, Fermat used variational methods to demonstrate that light follows the fastest route from one point to another, an idea which came to be known as Fermat’s principle, a cornerstone of geometrical optics. Variational Principles in Physics explains variational principles and charts their use throughout modern physics. The heart of the book is devoted to the analytical mechanics of Lagrange and Hamilton, the basic tools of any physicist. Prof. Basdevant also offers simple but rich first impressions of Einstein’s General Relativity, Feynman’s Quantum Mechanics, and more revealing and amazing interconnections between various fields of physics.

Variational Principles (Dover Books on Mathematics)

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Clear information packed on variation for physicists

Variational Principles (Dover Books on Mathematics)
B. L. Moiseiwitsch
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486438171

Book Description

This text shows how variational principles may be employed to determine the discrete eigenvalues for stationary state problems and to illustrate how to find the values of quantities that arise in the theory of scattering. Chapter-by-chapter treatment consists of analytical dynamics; optics, wave mechanics, and quantum mechanics; field equations; eigenvalue problems; and scattering theory. 1966 edition.

Customer Reviews:

Clear information packed on variation for physicists.......2006-01-13

This book from 1966 has been republished in a Dover edition and that usually means that it is a book that has become a classic, and this turns out to be correct in this case. There are many types of books on variational stuff. There are the mathematically inclined books on variational calculus. This is not one of them. This is a book on variational principles in theoretical physics. The style is clear and compact with many formulas and calculations given. It is a useful handbook on variational principles in physics and it could also serve as a text for a graduate level course. Its main weakness is perhaps in the lack of discussion of the conceptual background. It differs considerably from the classic by Yourgrau and Mandelstam which concentrates more on historical, philosophical, and conceptual aspects of the subject. But, if you have the right background for it, it is excellent.

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VARIATIONAL PRINCIPLES OF MECHANICS
Cornelius Lanczos
Manufacturer: University of Toronto Press
ProductGroup: Book
Binding: Hardcover
ASIN: B000KHGN72

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The Variational Principles Of Mechanics
Lanczos Cornelius
Manufacturer: University of Toronto Press
ProductGroup: Book
Binding: Hardcover
ASIN: B000I9VB88

Analytical Mechanics (Foundations of Engineering Mechanics)

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Analytical Mechanics (Foundations of Engineering Mechanics)
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Book Description

This is a translation of A.I. Lurie's classical Russian textbook on analytical mechanics. Part of it is based on courses formerly held by the author. It offers a consummate exposition of the subject of analytical mechanics through a deep analysis of its most fundamental concepts. The book has served as a desk text for at least two generations of researchers working in those fields where the Soviet Union accomplished the greatest technological breakthrough of the 20th century - a race into space. Those and other related fields continue to be intensively explored since then, and the book clearly demonstrates how the fundamental concepts of mechanics work in the context of up-to-date engineering problems. This book will help researchers and graduate students to acquire a deeper insight into analytical mechanics.

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