The variational principles of mechanics

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1. omnicognateOct 2025
   It's an old book and I can't vouch for it as I only just discovered it myself, but it appears to be very highly regarded, it focuses on precisely the questions you (and I) have, and just from the preface I like the author already [1]: The Variational Principles of Mechanics, by Cornelius Lanczos.

   There's a PDF here: https://pages.jh.edu/rrynasi1/PhysicalPrinciples/literature/...

   [1] An appetising quote:

   > The author is well aware that he could have shortened his exposition considerably, had he started directly with the Lagrangian equations of motion and then proceeded to Hamilton’s theory. This procedure would have been justified had the purpose of this book been primarily to familiarize the student with a certain formalism and technique in writing down the differential equations which govern a given dynamical problem, together with certain “recipes” which he might apply in order to solve them. But this is exactly what the author did not want to do. There is a tremendous treasure of philosophical meaning behind the great theories of Euler and Lagrange, and of Hamilton and Jacobi, which is completely smothered in a purely formalistic treatment, although it cannot fail to be a source of the greatest intellectual enjoyment to every mathematically-minded person. To give the student a chance to discover for himself the hidden beauty of these theories was one of the foremost intentions of the author.

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2. b215826Mar 2020
   > Other books like Introduction to Space Dynamics are very hand-wavey... Coordinate systems should not dictate the physics; it should be the other way around.

   It seems like you've not tried reading actual physics books written by real physicists and have only purveyed books known to a handful of engineering specialists, and have come to the conclusion that this is the best book for learning mechanics. I also fail to see why specialists wouldn't want to make use of methods that would make their lives easier and instead would want to dredge through pages of pointless algebra.

   > The rigorous approach to dealing with reference frames and coordinate systems is well applied to other fields like computer graphics

   In most universities around the world such things are taught as part of a standard vector calculus course or a mathematical methods course. You don't need this book (or any mechanics book for that matter) to learn such things.

   > Though I would be interested to read your textbook when it comes out

   I'll ignore the snideness of your comment, but will suggest that it's perhaps not a good idea to recommend/use books that take haphazard approaches for solving standard problems (some that were solve almost three centuries ago). In any case, my "textbook" would not look very different from the other 99% of textbooks written on analytical mechanics. If you want a recommendation, pick up Cornelius Lanczos's The Variational Principles of Mechanics [1], which is a real classic and a gem of a book.

   [1] https://www.amazon.com/Variational-Principles-Mechanics-Dove...

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3. b215826Jan 2020
   Among technical books, books by Cornelius Lanczos are some of the best (less popular) books I've read. Some quotes from his "The Variational Principles of Mechanics":

   From the Preface:

   Many of the scientific treatises of today are formulated in a half-mystical language, as though to impress the reader with the uncomfortable feeling that he is in the permanent presence of a superman. The present book is conceived in a humble spirit and is written for humble people.

   From Chapter 8:

   Put off thy shoes from off thy feet, for the place whereon thou standest is holy ground. -- EXODUS III, 5

   We have done considerable mountain climbing. Now we are in the rarefied atmosphere of theories of excessive beauty and we are nearing a high plateau on which geometry, optics, mechanics, and wave mechanics meet on common ground. Only concentrated thinking, and a considerable amount of recreation, will reveal the full beauty of our subject in which the last word has not yet been spoken.

   This book was also on Gerald Jay Sussman's must-read list of books [1]. Another great book of his is "Linear Differential Operators" -- if you've ever wanted an intuitive explanation for why d/dx is not Hermitian but d^2/dx^2 is, this is the book you need to read. A quote from the book that resonated with me when I first read it:

   Since the days of antiquity it has been the privilege of the mathematician to engrave his conclusions, expressed in a rarefied and esoteric language, upon the rocks of eternity. While this method is excellent for the codification of mathematical results, it is not so acceptable to the many addicts of mathematics, for whom the science of mathematics is not a logical game, but the language in which the physical universe speaks to us, and whose mastery is inevitable for the comprehension of natural phenomena.

   [1]: http://aurellem.org/thoughts/html/sussman-reading-list.html

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4. m31415Oct 2019
   For mathematics, I would recommend:

   1. "What Is Mathematics? An Elementary Approach to Ideas and Methods" by Courant and Robbins -- a general book on mathematics in the spirit of Feynman lectures.

   2. Strogatz's "Nonlinear Dynamics and Chaos" -- it's a bit narrow in scope (mostly dynamical systems with a little bit of chaos/fractals thrown in) but very good nonetheless.

   3. Tristan Needham, "Visual Complex Analysis", beautiful introduction to complex analysis.

   4. Cornelius Lanczos, "The Variational Principles of Mechanics" -- this is a physics book, but one of the classics in the subject, and as Gerald Sussman once remarked, you glean new insights each time you read it.

   5. Cornelius Lanczos, "Linear Differential Operators" -- an excellent treatment of differential operators, Green's functions, and other things that one encounters in infinite-dimensional vector spaces. This book has some very intuitive explanations, e.g., why d/dx is not self-adjoint (i.e., Hermitian), whereas d^2/dx^2 is.

   For chemistry, I would recommend "General Chemistry" by Linus Pauling, even though it's a bit outdated.

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5. selimthegrimOct 2016
   Another one in this vein is The Variational Principles of Mechanics by Cornelius Lanczos.
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6. tjlJul 2016
   There's two basic approaches to calculus of variations. One is based around the Gateaux Variation which basically is an extension of the normal derivative. So the vector x becomes x + epsilon*v and you take the limit as epsilon goes to zero. The second is due to Lagrange and uses the symbol lower case delta. The both amount to the same thing, but the notation is different. There's epsilon in the Lagrange approach, it's just buried.

   For Lagrange's approach, I recommend Lanczos' "The Variational Principles of Mechanics". It's published by Dover so if you can't find it in a library, it's only about $10 last I checked. For the other approach, we used John Troutman's "Variational Calculus and Optimal Control: Optimization with Elementary Convexity" in my Applied Math course on calculus of variations. It's good, but I don't think it's too approachable if you haven't had the lectures to go with it.

   I've done a lot of work with calculus of variations including working out the math for volume integration of variational gradients which is something that appears in beam deformation problems.

   One day I'll sit down and write books on calculus of variations, finite element analysis, and mechanics of deformable bodies. I was very unsatisfied with the texts I used for all of these in my undergrad and grad programs.

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