Customer Reviews

The Variational Principles of Mechanics (Dover Books on Physics)

5 star:		(17)
4 star:		(0)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

Surely Lanzcos is one of the few educators that compare to Feynman. This book presents an exceptionally lucid and engaging development of the main variational principles of mechanics with the discussion of Noethers theorem being the most cogent I've seen. While the problems may not be exceedingly difficult, this is because the problems are intended to illuminate the ideas under discussion.

Lanzcos divides mechanics very roughly into two camps, the vectorial/one-form viewpoint versus variational/analytical view. Since Lanzcos aspires to the variational view, the criticism concerning the lack of discussion of vectorial/one-form is rather moot and explains why there are not a great number of drawings. While this may be a problem for some, in my view figures can convey a false sense of understanding.

This is not a work that one can skim, it expects considerable reflection on the contents. We all differ in our needs and wants but in my view, this exposition exudes a truly infectious sense of wonder that such simple ideas could be so powerful and beautiful. But since no one book in this field may suffice for all one might also consider "Mechanics: From Newton's Laws to Deterministic Chaos" by Florian Scheck which admittedly covers mathematical methods that Lanzcos does not yet fails to capture the essence of the ideas as well as the Lanzcos text does.

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.

Lanczos' book is a compelling analysis of the principles of Lagrangian and Hamiltonian mechanics. It reminds me a bit of Feyman's Lectures on Physics because it focuses on the motivating principles behind advanced mechanics. In an elegant and flowing style, Lanczos guides the reader through a walking tour of the principles of mechanics, peppered with historical footnotes. If you understand how to use mechanics, but want to understand how the underlying principles are developed, this is an excellent choice.

Before reading this book, I knew almost nothing about analytical mechanics. My first text books taught Physics from a Newtonian approach, using mostly vectors and potentials. So, the first time I encountered Lagrangians and Hamiltonians I could not understand what these concepts meant. Because of that many areas of Theoretical Physics were forbidden for me: Phase and configuration space, Noether's theorem, Hilbert relativistic equations, Feynman quantum-mechanical interpretation of the principle of least action, and so on.

So, two years ago, I decided to buy this book. And what I encountered? A systematical and pedagogical approach to analytical mechanics, which enabled me to acquire the fundamentals of the subject.

For me, the most interesting features of this book are the following:

1) It explains the differences between VARIATION and DIFFERENTIATION, something that most books in the subject, leave behind.
2) It explains clearly D'Alembert Principle and the Principle of Virtual Work.
3) From those principles he derives the Principle of Least Action, using just elemental calculus.
4) He introduces the reader in Legendre's transformation and the relations between the two fundamental quantities of Analytical mechanics: Lagrangian and Hamiltonian.
5) You get the equations of movement corresponding to those quantities: Euler-Lagrange (Lagrangian) and canonical (Hamiltonian) equations.
6) A powerful insight in Configuration and Phase Spaces is given, including the wonderful Liouville's theorem.
7) Lanczos shows the analogies between Optics and Mechanics when he explains the Hamilton-Jabobi equations.

So, why to learn Analytical Mechanics and why to buy this book?? These are my reasons:

1) From a historical point of view, Analytical Mechanics was developed by Continental Mathematicians like Maupertuis, Euler, D'Alembert and Lagrange as a rival system to the Newtonian one exposed in the Principia Mathematica. Newton used vectors and potentials. Euler and Lagrange employed the Principle of Least Action.
2) It was Analytical Mechanics the first to develop the principle of energy conservation. Even when this principle in its general form was developed by Wilhelm von Helmholtz in 1847, the conservation of the sum of kinetic and potential energy was well known to Euler a century earlier.
3) The concept of phase space is very important in Thermodynamics. In fact, the definition of entropy given by Ludwig Boltzmann refers to the logarithm of a volume in phase space. Liouville theorem, which states the conservation of such phase space volumes, is very usefull today in black hole thermodynamics.
4) The quantum-mechanical interpretation of the Principle of Least Action given by Richard Feynmann was a fundamental contribution in the development of Quantum Field Theory, so any student who desires to progress in this field, must have substantial knowledge of Analytical Mechanics.

So, to all of you that eventually decide to buy this book, I wish you a good reading.

As another reviewer states, the book is somewhat broader than the title suggests. The chapters of the book are 1. The Basic Concepts of Analytical Mechanics 2. The Calculus of Variations 3. The Principle of Virtual Work 4. D'Alembert's Principle 5. The Langrangian Equations of Motion 6. The Canonical Equations of Motion 7. Canonical Transformations 8. The Partial Diffirential Equation of Hamilton-Jacobi 9. Relativistic Mechanics 10. Historical Survey 11. Mechanics of the Continua

There are a few examples and problems in the text, but I certainly wished for more to help illustrate concepts.

The author conveys in every chapter a strong reverence of the aesthetic value of the material and does an excellent job of showing how the themes of the analytic approach to clasical mechanics are preserved in relativity and quantum mechanics.

Cornelius Lanczos is the figure of paramount importance in the fields of Theoretical Physics, Applied Mathematics and Numerical Algorithms. In his lucid treatment of variational calculus Lanczos conscisely sums up the ideas that form foundation of the mathematical apparatus of contemporary fundamental physics. A must reading for any Graduate Student in Physics.

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. EDIT: Sept 2008. Recently I've found a textbook that clearly explains some of the fuzzy examples in Lanczos like varying the time: "Analytical Mechanics for Relativity and Quantum Mechanics" by Oliver Johns.

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.

I think this book is a wonderful reference for engineers taking a Lagrangian mechanics/dynamics course. This book dives a little deeper into the mathematics than most engineering oriented texts. It does so in a lucid fashion no less.

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.

Lanzcos has his name attached to many area of mathematical physics. Besides that, he is an excellent writer. This book on variationnal principle of mechanics is the most beautifull and profound one I've read on the subject. This book is well suited for undergrad or grad students, but even for reasearcher who uses the tools of analytical mechanics, it's worth reading it.