The Variational Principles of Mechanics (Dover Books on Physics)

In the spirit of the other reviewers here who appreciate this book’s elegance and erudition I cannot add much more. Lanczos artfully weaves a tapestry through not only the historical development of advanced mechanics but includes all of the technical tributaries as well.Digesting this book will make you a better engineer, a better educator, a better physicist and generally a better practicing (theoretical or experimental) scientist; without question.I spent the better part of a decade improving my understanding of general relativity and electrodynamics and I felt like I reach the pinnacle of understanding when I began to delve into Noether’s theorem, which Lanczos also covers briefly but builds up to it. It is unfortunate I was not well acquainted with Lagrangian physics, variational dynamics and the principle of least action before I was in my 40s, When I finally discovered this treasure trove of a physics book.As another reviewer admonished, the first two chapters are fantastic. They will guide you into the essentials, seducing a higher intellect you never evoked before.

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.

Lanczos' "Variational Principles of Mechanics" is an erudite piece of work that basically reconstructs the science of analytical mechanics bottom up, from the principle of virtual work to Einstein's equivalence principle and the origin of the gravitational redshift of spectral lines. The book contains very little material on the Newtonian, vector mechanics, being entirely devoted to the Lagrangian and Hamiltonian approaches to mechanics.The book provides a perfect introduction to the foundations of the all important principle of least action that pervades all of modern physics. In this regard, a nice companion to Lanczos' book is the treatise by W. Yourgrau and S. Mandelstam, "Variational Principles in Dynamics and Quantum Theory."One major "functionality" of Lanczos' book is to bridge the gap between our modern way of thinking and that of the classics. Anyone who has already tried to read the 1762 papers by Lagrange on Miscellanea Taurinensia or the 1834-1835 papers by Hamilton on the Phil. Trans. Royal Soc. knows that the classical literature is difficult to follow, part because of old-fashioned notation, part because we lack (well, at least I lack) the spirit of the times, making it difficult to understand some seemingly byzantine questions that the authors pose and ponder on in some of their (sometimes rather lenghty) writings. The book by Lanczos helps a lot both in terms of notation and ideas.I totally disagree with those that try to compare Lanczos' and Feynman's styles. Lanczos is seriously concerned with the history of the ideas, their evolution, and interpretations. There is nothing like that in Feynman's "The strange theory of light and matter," or in his "The character of physical law," and very little of it in his lectures on physics. Actually, Feynman pays very little credit to the historical development of the subjects. I am not saying that he personally does not recognize the credits, only that he does not communicate it. He admittedly has a "do it yourself" (or "did it myself") attitude towards physics, while Lanczos has a visible admiration for the greatness of his subject (without being cheesy) and is sensitive to its philosophical nuances and implications. I feel the difference between the two is like the difference between getting trained and getting educated.In summary: a must have in anyone's scientific library (unless you are an undergrad student cramming for your finals).