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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:

  1. Is there a text book you would recommend that phrases the constructions in classical mechanics via bundles without an appeal to transition functions?
  2. What are the drawbacks to this approach other than the fact that it makes computations less doable? (if it does that)
  3. 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?
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I just came upon this related question. See if any of those books works for you. –  Marek Dec 5 '10 at 10:06
    
coordinates are not as bad as you think: physics.stackexchange.com/questions/15002/… –  luksen Nov 29 '13 at 21:02
    
@luksen: I feel like coordinates are frequently something we impose on an object as opposed to something the object comes equipped with. I know that some people can make a lot of sense out of expressions involving coordinates, but I don't think I am one of them. –  Sean Tilson Dec 1 '13 at 15:43

8 Answers 8

up vote 15 down vote accepted

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.

  1. 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.

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

  3. It's a beautiful formalism.

  4. 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.

  5. 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).

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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.

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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.

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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.

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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.

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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.

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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.

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