A good reference for the tangent bundle formulation of the Lagrangian

Question

I wrote a small section for the wikipedia article about Lagrangian mechanics, that is intended to be a short and easy but formal introduction to the subject for mathematicians (the current version of the article is supposed to be introductory but it is very confusing for mathematicians). It goes basically like this: the configuration space is a Riemannian manifold, which in the simplest case is just $Q=(\mathbb{R}^3)^k$ or a sub-manifold of that, and the Lagrangian is a function defined over the tangent bundle of $Q$. Typically (again, in the simplest case), it is defined as the difference between a kinetic energy function, which is, well, you know what it is, and a potential function which is constant on the fibers.

Anyway, I wrote something of this flavor and it was deleted because I didn't give a source for this formulation. So my question is: can you name a good, concise, clear, hopefully openly available source that defines the Lagrangian in this way and would be accepted by Wikipedia?

Answer 1

You just seem to be writing an “introduction” to the geometric aspects of the theory, so I don’t see anything wrong with quoting a more authoritative/advanced source (afterall what’s the point of academic articles on wikipedia if you’re not simplifying/summarizing in some regard a more complete/advanced source). But here are some textbooks which I liked reading (in no particular order)

• Abraham and Marsden: Foundations of Mechanics, available on Caltech’s library website, sections 3.5, 3.7,3.8 specifically. This is the “gold standard” for mechanics for mathematicians.
• Curtis and Miller: Differential Geometry and Theoretical Physics, Chapters 1,2,7. Chapters 1 and 2 are well… clearly introductions… and I think it’s a nice summary to get mathematicians into the mindset of some basic Physics. Also, chapter 7 includes a nice introduction to holonomic systems, along with some helpful occasional commentary relating these concepts to how Goldstein describes things in his book. Along the way, the reader is introduced to differential forms as well (to discuss the Hamiltonian formulation). Actually the entire book shows how nicely differential geometric concepts are used in Physics (Lagrangians, Hamiltonians, rigid bodies, Principal bundles and basic gauge theory etc)
• Loomis and Sternberg: Advanced Calculus, from Sternberg’s website, chapter 13. The 3-page preface to the chapter itself is extremely enlightening for math students getting to physics. The only downside is it starts off with Hamiltonian mechanics, and only later we have Lagrangian mechanics (section 13.8).
• Michael Spivak: Physics for Mathematicians, Mechanics I, chapter 12. Spivak takes a great deal of effort to explain the concise physics notation in a manner that is appreciable for students comfortable with differential geometry (at the basic level of his Vol I book). But perhaps this isn’t what you’re after since he doesn’t go axiomatically like “start with a semi-Riemannian manifold $M$ and a smooth function $L:TM\to\Bbb{R}$…”, but rather tries to link it back to his earlier chapter where he discussed d’Alembert’s principle.
• V.I. Arnold: Mathematical Methods of Classical Mechanics, chapter 4, section 19 in particular. Arnold’s book is of course a classic and considered one of the “gold standards” (not my personal favourite though since it seemed to be too scattered and brief with details).
• Godinho and Natário: An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity, chapter 5 (5.1 specifically). This is a very readable chapter for an individual who knows the basics of manifolds and what a Riemannian metric is (and slightly later on, what the Levi-Civita connection is… but all of this is covered earlier in the book). This book clearly outlines the various ingredients of the theory (and also nicely discusses constraints).
• C. Godbillon: Géométrie différentielle et mécanique analytique, chapter XI. I can’t read French, but I can make roughly make out the contents of this last short chapter (~10 pages), and it seems very clear about the mathematical structure of things (as expected from a French book). I’ve also heard that this is a very well-regarded book (perhaps due to its concise nature), and it’s also cited in Abraham and Marsden’s book.
• Cortés and Haupt: Mathematical Methods of Classical Physics, chapter 2. This is a very readable chapter which immediately gets into the Lagrangian formulation (also a great book all around). On page 5-6 (Definition 2.1 and Example 2.3 specifically) already has everything I’m guessing you’ll need to cite on Wikipedia.

As far as citing your introductory claims on Wikipedia, I’d go with Cortés and Haupt, or Curtis and Miller, or Godinho and Natário (even though Abraham and Marsden is the most thorough).