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I would like to know: what are some of the best introductory books to Lagrangian Mechanics?

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Rather than a book, consider the Susskind Physics lectures made available on Youtube as well as Stanford on iTunes via iTunes-University. You want to watch the latest version of the course lectures on Classical Mechanics. After a small bit of introductory material, Susskind gets right into the Lagrangian and Hamiltonian approach to mechanics using a simple approach for easy understanding. Better yet, Susskind also covers topics like Liouville's Theorem, Poisson Brackets, and examples of applying Lagrangian methods to Maxwell's equations for Electromagnetic Field.

I almost forgot. Susskind's lectures are available on-line via the website located at: http://theoreticalminimum.com/home. Take that link, jump to the menu item Courses, and begin watching the lectures on Classical Mechanics. Each lecture is roughly 2 hours long, entertaining, and at a reasonably descent level consistent with advanced undergraduate courses in Classical Mechanics.

Susskind also has a book that parallels this particular video series called "The Theoretical Minimum". I believe that this name, "The Theoretical Minimum" comes from the Russian physicist L.D. Landau who preferred to welcome students into his graduate programs who had already demonstrated knowledge of the theoretical minimum amount of physics to succeed in his graduate PhD programs.

By the way, the book Mechanics, by L.D. Landau and E.M. Lifshitz is excellent too but more towards a graduate level of mechanics. It starts out with Lagrangian methods in chapter 1 (generalized coordinates, principle of least action, Lagrangian, Euler-Lagrange equation, etc.).

There are many other book recommendations that can be found here on Physics S.E. but I do believe that the best starting place is Susskind's lectures.

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  • $\begingroup$ As I understand it, "The Theoretical Minimum" is popularization of mechanics and not a textbook treatment of the subject in the sense of Taylor, Goldstein, K&K and so on. $\endgroup$
    – Kyle Kanos
    Commented Jun 25, 2015 at 20:05
  • $\begingroup$ @KyleKanos I recommend the book, and he does stuff in there such as deriving [truly, not in a handwavy sense] the E.L. eqs from the least action principle (in the same way Euler did, it turns out), finding the rate of precession given angular momentum and an external torque, and plenty of other good math stuff. So it's far from popular. It does lack in problems a bit though. $\endgroup$
    – user12029
    Commented Jun 25, 2015 at 20:20
  • $\begingroup$ This is a very strong second for "The Theoretical Minimum". It's not quite a textbook but it's far more than a popularization, and there's definitely enough detail to clarify exactly how the theory works and to get you started solving problems. And it's beautifully written. $\endgroup$
    – WillO
    Commented Jun 25, 2015 at 20:21
  • $\begingroup$ @NeuroFuzzy: true story: we had a kid in chat a few months ago claiming he knew CM well because of Susskind's book & online course. Through conversation, turns out it was a sham & he didn't actually learn anything b/c the book doesn't go into details that it should and ignores problem solving aspect. Data point of 1, but still telling to me that it's not worth the $15 compared to others. $\endgroup$
    – Kyle Kanos
    Commented Jun 25, 2015 at 20:28
  • $\begingroup$ @KyleKanos I am not sure of the OP's motivation to learn Classical Mechanics from the Lagrangian approach but like many of us, it is because of passion to learn, lost opportunity from when we were younger, or other things; and, not necessarily in preparation for a degreed education. For pursuit of BS Physics or graduate schools, yes good text books (as you suggested) are desired and important. For for the weekend hobbyist (like myself and others), the Susskind approach is good enough or a good enough start. $\endgroup$
    – K7PEH
    Commented Jun 25, 2015 at 20:55