Classical mechanics without coordinates book I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think of the fields that arise in physics as sections of vector bundles (or maybe principal bundles) and would love an approach to classical mechanics or what have you that took advantage of this.
Now for the questions:


*

*Is there a text book you would recommend that phrases the constructions in classical mechanics via bundles without an appeal to transition functions?  

*What are the drawbacks to this approach other than the fact that it makes computations less doable? (if it does that)  

*Are there benefits to thinking about things this way, ie would it be of benefit to someone attempting to learn this material to do it this way?

 A: To answer 3: it really depends on why you want to learn this material.  To me, the modern view is important because it is very elegant and generalizable (you can "do" classical mechanics on any Poisson manifold).  It also leads to very interesting mathematics.  For example, the evolution of an observable is given by $f' = \{H,f\}$ where $H$ is the Hamiltonian function and $\{,\}$ is the Poisson bracket.  In the Heisenberg picture of quantum mechanics, an observable (represented by an operator $A$) evolves according to $A' = [H,A]$ where $H$ is the quantum mechanical Hamiltonian and $[,]$ is the Lie bracket (commutator).  This resemblance has led to things like deformation theory and quantization.  
A: I recommend a recent book by Leon Takhtajan, "Quantum mechanics for mathematicians". It starts with an introduction to classical mechanics aimed at mathematicians and explains the coordinate-free approach, among other things.
A: I think that Geometric Algebra suits Classical Mech without Coords, and more.
There are a lot of free resources in the net. 
The approach is computational easy.
from the Hestenes book New Foundations for Classical Mechanics
quoting:  

...introduction to geometric algebra
  as a unified language for physics and
  mathematics... introduces new,
  coordinate-free methods for rotational dynamics and orbital
  mechanics, developing these subjects
  to a level well beyond that of other
  textbooks. These methods have been
  widely applied in recent years to
  biomechanics and robotics, to computer
  vision and geometric design, to
  orbital mechanics in governmental and
  industrial space programs, as well as
  to other branches of physics. The book
  applies them to the major
  perturbations in the solar system,...

or Geometric Algebra and its Application to Mathematical Physics by Chris J. L. Doran (Chris Thesis) free download
....
or Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics (GA)
free download  

... This has produced a
  comprehensive language called
  Geometric Algebra, which I introduce
  with emphasis on how it simplifies and
  integrates classical and quantum
  physics.
  ... After explaining the utter
  simplicity of the GA grammar ..
  unique features of the mathematical
  language: 
  (1) GA seamlessly integrates
  the properties of vectors and complex
  numbers to enable a completely
  coordinate-free treatment of 2D
  physics.
  (2) GA articulates
  seamlessly with standard vector
  algebra to enable easy contact with
  standard literature and mathematical
  methods.
  (3) GA Reduces “grad, div,
  curl and all that” to a single vector
  derivative that, among other things,
  combines the standard set of four
  Maxwell equations into a single
  equation and provides new methods to
  solve it.
  (4) The GA formulation of
  spinors facilitates the treatment of
  rotations and rotational dynamics in
  both classical and quantum mechanics
  without coordinates or matrices.
  (5) GA provides fresh insights into the
  geometric structure of quantum
  mechanics with implications for its
  physical interpretation.
  All of this generalizes smoothly to a completely coordinate-free language for
  spacetime physics and general
  relativity to be introduced in
  subsequent papers.

A: The book 

Marsden and Ratiu, Introduction to mechanics and symmetry
presents classical mechanics from a modern differential geometry point of view.
Although nothing for beginners, it is unique in presenting a point of view in which all classical conservative systems (including those of field theory) are presented in a Hamiltonian framework.
A: 1.
I am in love with Fecko's Differential Geometry and Lie Groups for Physicists. Despite not being just about mechanics (but rather about more or less all rudimentary modern theoretical physics) it discusses both Lagrangian and Hamiltonian formalism. It also provides countless exercises (with nice hints) so that you can really get a feel for the matter.
2.
I can't think of any major drawbacks. Of course, if the problem has no symmetry you sometimes have no other choice than to go back to some coordinates and solve numerically. But this is probably non-issue for you because I suppose you first want to understand physical problems with some structure.
3.
There are countless benefits. To list just few of them.

*

*relation to symmetries and conserved quantities becomes obvious. Noether's theorem in Hamiltonian formalism is so amazingly simple statement (Hamiltonian is constant for symmetry flow if an only if the generator of the symmetry is constant for Hamiltonian flow) that one has to wonder where all the long-winded coordinate calculations went.


*Not only are the calculations short, one also gains valuable geometrical insights e.g. about the flow of the configuration on the manifold.


*It's a beautiful formalism.


*I don't know about others but whenever I have to calculate in coordinates I become nervous. I can compute the results but after few pages when most of the quantities mysteriously cancel, you don't really know why what you derived is true. So then you go back to geometry and lo and behold, the derivation is just few lines and obvious. Of course I am exaggerating now but that's the way I feel.


*It's the basis for all of modern physics. If the above four points were true in classical mechanics, they are even more true when dealing with things like gauge theories (and that is where the full beauty and power of mathematics comes out).
A: One classic book along these lines is

Mathematical Methods of Classical Mechanics. V. I. Arnold. Graduate Texts in Mathematics vol. 60, Springer, New York, 2000. Available e.g. here.

This book is mathematically very formal and very clear; I loved it when I took analytical mechanics because it avoids the phycisists' smudges of rigour and presents one clear, coherent structure. He does not start with a handed-down Big Principle (as in, these are Hamilton's equations in symplectic form and let's see how one can construct experimental mechanics from them), but he does formulate the basic theory very cleanly, and from there he moves higher up in abstraction. 
A: The story is not very long.  A symplectic manifold is a manifold $X$ with a nondegenerate, closed two-form, $\omega$.  Given this, from a function $h$ ("Hamiltonian") we can construct a vector field $v$ by $\omega(v,-) = dh$.
Flow by $v$ defines the classical trajectories.  (Note we have not used a metric on $X$, i.e. $v$ is NOT the gradient of $h$.)  In fact, $h$ remains constant (conserved) in the flow, since $v(h) = \omega(v,v) = 0.$  Also note ${\mathcal L}_v \omega = 0$, meaning $\omega$ is conserved in the flow -- in particular, so is the Liouville phase $\omega^{\wedge n}$. 
For an example, take $X$ to be the cotangent space $T^*({\mathbb R}^n)$ with coordinates $x$ (position) and $y$ (momentum), with $\omega = {\rm d}x \wedge {\rm d}y$.  Take $h = y^2/2 + V(x)$, i.e. KE + PE.  Then $v = y {\partial \over \partial x} - V' {\partial \over \partial y}$, so the flow equations are $\dot{x} = y$, $\dot{y} = -V'$, or $\ddot{x} = -V'$, Newton's law.
A: This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that do not yet have commentary.



*

*W. M. Oliva. Geometric mechanics (Lecture Notes in Mathematics,
Vol. 1798. Springer, Berlin, 2002).

*J. José and E. J. Saletan. Classical dynamics: a contemporary approach (Cambridge University Press, Cambridge, 1998). This takes an approach based on tangent bundles.
