Any good resources for Lagrangian and Hamiltonian Dynamics?

Question

I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics.

So far at my university library, I have found many books on both subjects, but not ones with good practice questions and answers. I have Schuam's outline of Lagrangian Dynamics, but didn't really find a lot of practice questions.

Any suggestions would be greatly appreciated!

Answer 1

I'll write here a list of my personal favorites plus some commonly used books.

I wouldn't be surprised if your teacher chose either one of the books below as a textbook:

i) Mechanics, the first volume of the Landau course on Theoretical Physics;

ii) Goldstein's book "Classical Mechanics";

iii) Taylor's book "Classical Mechanics";

iv) Marion's book "Classical Dynamics of Particles and Systems";

v) Symon's book "Mechanics";

Goldstein's book may be very appropriate for a first or second course on the topic, but I don't believe it displays a very formal approach to the subject. I'd suggest it to someone who's not interested in the mathematical structure of Mechanics. Even though, good for a starter.

Taylor's book has some very good exercises, but the book itself does not please me at all since it's informal, prolix and severely incomplete in most topics. Same goes to Marion's book, and even though Symon's is a little bit better, it didn't please me either.

The best book in this list if definitely Landau's, but I don't find it as good as most people picture it. I didn't read the whole Landau series (not even half, actually), but until now it's the worst of them all, for me. It still carries much of the author's incredible insights and some very nice solved exercises, but (as Arnol'd pointed out) there are a some mistakes and fake demonstrations on the book. Don't trust all of his "proofs" and you'll be safe.

Now I'll point some books that really helped me throughout my studies:

Arnol'd's "Mathematical Methods of Classical Mechanics": This book is simply the best book you can get your hands on after acquiring familiarity with the subject (after a first course using Goldstein's or Landau's book, for example). It's thorough, the maths are just clear and not extravagant, the proofs are very simple and you can get some contact with phase space structures, Lie algebras, differential geometry, exterior algebra and perturbation methods. Arnol'd's way of writing is incredibly clean, as if he really wanted to write a book with no "mysteries" and "conclusions that jump out of nowhere". The exercises are not very suited for a course.

Saletan's "Classical Dynamics: a Contemporary Approach": Very nice book. A little more developed mathematically than Arnol'd's, since it delves into the structure of the cotangent bundle and spends a great deal of the book talking about chaos and Hamilton-Jacobi theory. The proofs are not very elegant, but I'd chose it as a textbook for a graduate course. Some nice exercises.

Fasano's "Analytical Dynamics": Also a graduate-textbook-style one. Very close to Saletan's way of writing, trying to explain to physicists the mathematical nature of Mechanics without too much rigor, but developing proofs of many theorems. Very nice chapter of angular momentum, very nice exercises (some of them, solved!). Incredibly nice introduction to Lie derivatives and canonical transformations, and very philosophically inclined chapters so to answer "why is this this way" or "what does that mean, really?".

Lanczos' "The Variational Principles of Mechanics": This book is kept close at all times. Not suited (at all) as a textbook, more like a companion throughout life. The most philosophical, inquiring and historical Mechanics book ever written. If you want to read a very beautiful account on the the structure, the problems, the development and the birth of mechanical concepts I'd recommend this book without blinking. It is a physics book: calculus and stuff, but looks like it were written by someone who liked to ask deep questions of the kind "why do we use this instead of this, and why is mathematics such a perfect language for physics?". It's just amazing.

Marsden's "Foundations of Mechanics": This is the bible of Mechanics. Since it's a bible, no one ever read it all or understood it all. Not to be used as textbook ever. It's a book aimed for mathematicians, but the mathematical physicist will learn a lot from it, since it's quite self contained in what touches the maths: they're all developed in the first two chapters. Even though, very acidly developed. Hard to read, hard to understand, hard to grasp some proofs... In general, hard to use. Even though, I really like some parts of if... A lot.

Ana Cannas' "Introduction to Symplectic and Hamiltonian Geometry": Another mathematics book, but this is the best one (in my humble opinion). Can be found for free (in English) at www.impa.br/opencms/pt/biblioteca/pm/PM_11.pdf .

Kotkin's "Collection of Problems in Classical Mechanics": Last but not least, filling in the "with a lot of exercises" hole, Serbo & Kotkin's book is simply the key to score 101 out of 100 in any Mechanics exam. Hundreds of incredible, beautiful, well thought problems together with all (ALL!) their solutions at the end. From very simple to "hell no I'm not trying this one" problems, this book should be a reference to everyone studying the subject. Some of the problems are so nice that you can even publish notes in teaching journals about them, like I've seen once of twice before.

Well, this is my humble contribution. I hope it helps you!

EDIT.: I just noticed I forgot one book that really changed my life: Spivak's "Physics for Mathematicians, Volume I: Mechanics". The physicist should not be scared about the title. This is the best book ever written about Mechanics. I actually have plans of taking vacations only to read it all. There's nothing missing, all the mathematics is rigorous and perfect, and there's not a single step that isn't clarified by the author (who said he was learning Mechanics himself whilst writing this book). There are moments he pauses to inquire about contact structures in symplectic manifolds, but also moments where the reason for inquiry is the fact that forces are represented by vectors; and then he goes back to Newton's time where vectors didn't exist... And tries to explain how people used to see forces and momentum at the time, in his opinion. It's just magical. He's as worried about presenting the content of the subject as to try to grasp why the definitions are the way they are, and then justify it historically. Sorry if I'm being redundant, but please read this book!

Answer 2

Edward A. Desloge Classical Mechanics, Vols I and II. Wiley-Interscience, 1982. The 93 chapters are remarkably short. This highly systematized and detailed book includes plenty of examples and several hundred problems, most with answers. From Newtonian mechanics it progresses to Lagrangians, Hamiltonians, and touches upon Relativity. Curiously, it is not as popular and talked about as it deserves. These two volumes belong to the bookshelf of anyone interested in learning or teaching the subject. The question was originally asked five years ago apparently by a student, perhaps today a PhD in Physics. But this reference may be useful for other students of CM.

The two volumes can be downloaded in djvu format here http://gen.lib.rus.ec/search.php?req=Desloge&lg_topic=libgen&open=0&view=simple&res=25&phrase=1&column=def

Answer 3

Read Lagrange's Mécanique analytique (English translation: Analytical Mechanics). The book is split up into two parts: statics and dynamics. The first chapter, "The Various Principles of Statics," is a beautiful historical overview. Lagrange works out many problems; for example, he has a chapter entitled "The Solution of Various Problems of Statics." But, since you're interested in dynamics, you might want to focus on the second part of Analytical Mechanics.

Answer 4

Ok, so some of the other answers here seem like they may have forgotten what it's like to be in your position as a starting student (Goldstein as a starter book... really? Lagrange's original work?! Did you even read the question?). There are a number of great books perfectly suited to your exact position. Here are some books I think deserve consideration (I'm adding this because this question is on the community wiki, so hopefully this answer will be beneficial to at least one person in the future.):

• Taylor's Classical Mechanics. This is the canonical book for undergraduate classical mechanics. It is used almost everywhere in the US for their junior-level undergraduate classical mechanics class, as it's clear, has lots of good problems, and doesn't take mathematical background for granted. It is a great book for a starting course in the subject. A solutions manual for it can be found on the web.
• Helliwell and Sahakian's Modern Classical Mechanics. This book is like Taylor, but clearer, as well as covering more topics at a higher level. It has a number of great problems, for which you can find a solutions manual to online. Another massive strength of this book is the sheer number of worked examples; without exaggeration, most chapters have ten contiguous pages of examples. These massively help kickstart the learning process, as you get to see how to apply the theory right away. I think this is probably the best book for the dedicated student going through their first upper-undergraduate course on classical mechanics.

We are now getting into the land of "second-course" books. These books may be suitable to a first course in classical mechanics, but they will probably be better as your second or third book on classical mechanics.

• Landau and Lifshitz's Mechanics. This is the first volume in the legendary Theoretical Minimum of Landau. It is a short book, on the order of 150 pages. As you might guess, it is incredibly potent stuff. Landau cuts all of the fat out of classical mechanics, leaving you with gems of insight throughout. However, Landau's brevity can also be confusing to the beginner, as he make bold claims that require pages of exposition for their nuance to be unraveled. This is why I think this book is better as a second reading, as you will be able to fill in everything that Landau cuts out. It contains ~10 problems a chapter on average, all with full solutions.
• Goldstein's Classical Mechanics. This is the canonical graduate classical mechanics book. It has been around for a long time. It is good; the exposition is clear, and it covers pretty much all of the topics you would want to see in a graduate course. It even gets to field Lagrangians in the latter chapters, an important bridge to the next course you would take in graduate school, classical electrodynamics. Its problems are very good, and many of them are quite difficult. They are great for preparing for comprehensive exams for PhD candidacy. The normal problem level for this book is comparable to the three star problems in Helliwell and Sahakian, so it's a definite jump up in difficulty.
• Arnol'd's Mathematical Methods of Classical Mechanics. This is the "advanced mechanics" book, for use after a course in Goldstein/Landau. Its mathematical level is much higher than either of Goldstein or Landau, and it's proof-based. It largely covers things geometrically, as opposed to the mostly coordinates-based approach of all of the books we've talked about so far. It has a number of good problems, with solutions/answers to some right after the question. Its appendices are one of the best parts of the book; they cover all sorts of neat math that you would use in advanced classical mechanics. It's a great book for the more mathematically curious/inclined.

Here are some quick-fire extra resources that you may want to check out. I will offer less explanation of them. They are roughly organized by level.

• Morin's Introduction to Classical Mechanics. This book is meant for honors freshman classical mechanics at Harvard. It covers Newtonian mechanics very thoroughly, and some of its problems are quite difficult. It covers Lagrangians for a chapter, as well as central motion and rigid bodies. In my opinion, its coverage is not sufficient to actually understand these subjects at an upper-undergraduate level. If your Newtonian mechanics is rusty, though, this book would be a solid choice to read before going into the books listed above.
• Kevin Zhou's handouts. These aren't on Lagrangian/Hamiltonian mechanics, but they are quite fun. They are handouts made for students prepping for the International Physics Olympiad, so all of the problems are tricky, and are meant to teach important ideas in the big four mechanics of physics. They cover roughly the first two years of physics subject matter at top schools. If you want to have some hard practice with Newtonian mechanics, these would be a great resource. (I just had to add this resource; doing these handouts has been a lot of fun for me, so I wanted to share).
• David Tong's notes on classical dynamics. These are at the level of the advanced undergraduate. They would be a nice supplement to a course in Taylor or Helliwell and Sahakian.
• Kotkin and Serbo's Exploring Classical Mechanics. This is a great problem book for classical mechanics. Sometimes with undergraduate books, you can start to get slightly annoyed with the lack of truly tricky problems; this will fill that need. It has great problems, many of them quite difficult.
• Gupta's Classical Mechanics of Particles and Rigid Bodies. This book is a hidden gem. The best way to describe it is Landau with more exposition, and way more problems (with examples too!). It is quite a good book, but also somewhat rare from what I've seen.
• Jose and Saletan's Classical Dynamics: A Contemporary Approach. This book is somewhere on the order of Goldstein, but more mathematical (gets into tangent bundles), and also clearer in places. It has a number of great problems, and there are solutions to them online. I think this book may be a good choice if you want a more mathematical version of Helliwell and Sahakian.
• Maxim Jeffs' notes on classical mechanics and symplectic geometry. These notes are another hidden gem. The exposition is clear, and the problems in there are quite good (and classic). For the more mathematically inclined.

There are way more books on classical mechanics than what I have listed, but these are some of the best in my opinion. If you start with these, you will be ready to read any classical mechanics book you want to (barring some advanced books that are basically just symplectic geometry).