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October 19, 2006

A Course on Classical Mechanics

Posted by John Baez

Here’s a course on classical mechanics for people who can handle some differential geometry:

This may come in handy if you’re following this Fall’s course on quantization and cohomology - so far we’re mostly covering the same ground at high speed, but soon we’ll shoot ahead.

These are some course notes and homework problems for a mathematics graduate course on classical mechanics. I’ve taught this course twice recently. The first time I focused on the Hamiltonian approach. This time I started with the Lagrangian approach and derived the Hamiltonian approach from that.

My goal was to understand the principle of least action from more viewpoints than I had before, including some historically important ones that you don’t see much these days. I also wanted to show how classical mechanics foreshadows all the ideas of mechanics.

Derek Wise took notes. Here they are:

  • Week 1  (Mar. 28, 30, Apr. 1) -The Lagrangian approach to classical mechanics: deriving F = ma from the requirement that the particle’s path be a critical point of the action. The prehistory of the Lagrangian approach: D’Alembert’s "principle of least energy" in statics, Fermat’s "principle of least time" in optics, and how D’Alembert generalized his principle from statics to dynamics using the concept of "inertia force".
  • Week 2  (Apr. 4, 6, 8) - Deriving the Euler-Lagrange equations for a particle on an arbitrary manifold. Generalized momentum and force. Noether’s theorem on conserved quantities coming from symmetries. Examples of conserved quantities: energy, momentum and angular momentum.
  • Week 3  (Apr. 11, 13, 15) - Example problems: 1) The Atwood machine. 2) A frictionless mass on a table attached to a string threaded through a hole in the table, with a mass hanging on the string. 3) A special-relativistic free particle: two Lagrangians, one with reparametrization invariance as a gauge symmetry. 4) A special-relativistic charged particle in an electromagnetic field.
    • A Spring in Imaginary Time. Learn how replacing time by "imaginary time" in Lagrangian mechanics turns dynamics problems involving a point particle into statics problems involving a spring.
    • Answers by Garett Leskowitz.
    • Answers by Jeffrey Morton.
    • Answers by Erin Pearse.
    • Answers by Alex Hoffnung.
  • Week 4  (Apr. 18, 20, 22) - More example problems: 4) A special-relativistic charged particle in an electromagnetic field in special relativity, continued. 5) A general-relativistic free particle.
  • Week 5  (Apr. 25, 27, 29) - How Jacobi unified Fermat’s principle of least time and Lagrange’s principle of least action by seeing the classical mechanics of a particle in a potential as a special case of optics with a position-dependent index of refraction. The ubiquity of geodesic motion. Kaluza-Klein theory. From Lagrangians to Hamiltonians.
  • Week 6  (May 2, 4, 6) - From Lagrangians to Hamiltonians, continued. Regular and strongly regular Lagrangians. The cotangent bundle as phase space. Hamilton’s equations. Getting Hamilton’s equations directly from a least action principle.
  • Week 7  (May 9, 11, 13) - Waves versus particles: the Hamilton-Jacobi equation. Hamilton’s principal function and extended phase space. How the Hamilton-Jacobi equation foreshadows quantum mechanics.
  • Week 8  (May 16, 18, 20) - Towards symplectic geometry. The canonical 1-form and the symplectic 2-form on the cotangent bundle. Hamilton’s equations on a symplectic manifold. Darboux’s theorem.
  • Week 9  (May 23, 25, 27) - Poisson brackets. The Schrödinger picture versus the Heisenberg picture in classical mechanics. The Hamiltonian version of Noether’s theorem. Poisson algebras and Poisson manifolds. A Poisson manifold that is not symplectic. Liouville’s theorem. Weil’s formula.
  • Week 10  (June 1, 3, 5) - A taste of geometric quantization. Kähler manifolds.

You can find errata for these notes here. If you find more errors, please email me!

Here are some homework problems from the last time I taught the course.

Again, please let me know if you spot errors.

You can also download TeX or LaTeX files of the homework problems and some solutions, if for some bizarre reason you want them. However, the authors keep all rights to this work, except when stated otherwise.

Posted at October 19, 2006 1:47 AM UTC

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Re: A Course on Classical Mechanics

I’ve looked at some of these notes before and they’re great.

I would like to make some comments about Kaluza-Klein theory, which you discussed in the Week 5 lecture.

The first time I learnt of this beautiful fact (that the path an electromagnetic particle follows is the projection of a geodesic on the bundle) was in the excellent book:

I’d like to ask: can one think of string dynamics and gerbes in this way?

Of the many things an (abelian) gerbe-with-connection is, I understand that it is at least a line-bundle-with-connection on the loop space.

In addition, if your manifold has a metric (as any self-respecting manifold has), then supposedly this can be lifted to loop space.

So - lets play the Kaluza-Klen game… on loop space! We get a metric on the line-bundle-over-loop-space, and we project the geodesics down to loop space, to obtain dynamics of the loops.

What kind of dynamics do we get out?

Posted by: Bruce Bartlett on October 19, 2006 1:58 PM | Permalink | Reply to this

Re: A Course on Classical Mechanics

So - lets play the Kaluza-Klein game… on loop space! […]

What kind of dynamics do we get out?

Beautiful question.

One way to see the Kaluza-Klein mechanism is to consider the Laplace(-Beltrami) operator

(1)Δ \Delta

which locally reads

(2)Δ=(d+d ) 2g ij( i jΓ k ij k)+, \Delta = (d + d^\dagger)^2 \propto g^{ij} (\partial_i \partial_j - \Gamma^k{}_{ij}\partial_k ) + \cdots \,,

insert the metric on X×S 1X \times S^1 and study the field equation

(3)Δψ= \Delta \psi = \cdots

of a given particle.

Now, it is true that the dynamics of the (Nambu-Goto/Polyakov) string is essentially given by the same kind of wave equation, too - but on loop space. (You can find some references collected here.)

While I haven’t checked it, I would not be surprised if (modulo the usual technical issues with differential geometry on loop space) the KK-mechanism goes through here just as well and produces the Kalb-Ramond field.

If one allows oneself to work naively with locally coordinatized versions of Δ\Delta on loop space, it should be a simple matter to check this.

(But I am afraid even if I had the time to start this computation right now, you would come up with yet another deep question before I finished. :-)

Said that, I should remark that from the perspective of physical applications, KK-theory on loop space seems (unless I am missing something) much less compelling than KK-theory on target space.

But something closely related is of considerable interest:

what is the KK-reduction of gravity coupled to a 2-gerbe on a circle bundle?

(The issue is: what is it in terms of globally defined quantities? We know what it is in terms of local data of a trivial 2-gerbe.)

Posted by: urs on October 19, 2006 3:11 PM | Permalink | Reply to this

Re: A Course on Classical Mechanics

Bruce Bartlett writes:

In addition, if your manifold has a metric (as any self-respecting manifold has), then supposedly this can be lifted to loop space.

Sure. A tangent vector in the space of loops in XX is a loop of tangent vectors in XX. If we have two of these, say

v,w:S 1Xv, w : S^1 \to X

there’s an obvious way to define their inner product using a Riemannian metric on XX, namely

v,w= S 1v(t),w(t)dt\langle v, w \rangle = \int_{S^1} \langle v(t), w(t) \rangle d t

So, if XX is a Riemannian manifold, the free loop space LXL X becomes a Riemannian manifold as well. Of course, it’s infinite-dimensional. So, we should be precise: if we use smooth loops, it’s a Riemannian Fréchet manifold.

None of this used anything special about the circle S 1S^1 except that we can do integrals of smooth functions on it and be sure they converge. So, we could replace S 1S^1 by any compact manifold Σ\Sigma with a volume form!

Theorem: If Σ\Sigma and XX are smooth finite-dimensional manifolds, Σ\Sigma is compact and equipped with a volume form, and XX is equipped with a Riemannian metric, then the space of smooth maps C (Σ,X)C^\infty(\Sigma, X) is a Riemannian Fréchet manifold.

There is, however, something special about using the circle for Σ\Sigma. Brylinski explains it in his book on gerbes.

In this case, the space of smooth maps C (Σ,X)=LXC^\infty(\Sigma, X)= L X is not just a Riemannian manifold. It’s a Kähler manifold!

In other words, it gets a complex structure and a symplectic structure which get along nicely with each other and the Riemannian structure.

As discussed in the week 10 lectures of this course on classical mechanics, Kähler manifolds are just what you want for geometric quantization.

So, there’s something espeically nice going on.

And I hope Urs sees why this is so tantalizing. I’d like to know the analogue of geometric quantization for gerbes. Normally geometric quantization wants an integral Kähler structure, which in turn gives a line bundle; then we take the space of holomorphic sections to be our Hilbert space. I don’t know the analogue of all that for gerbes.

But, a Riemannian metric on XX gives a Kähler structure on LXL X! And, a gerbe on XX gives a line bundle up on LXL X! So, a gerbe on a Riemannian manifold seems in some ways like the right analogue of a line bundle on a Kähler manifold.

But the analogy is weird: I know how to obtain a line bundle (up to isomorphism) from an integral Kähler manifold, but I don’t know how to obtain a gerbe (up to equivalence) from an integral Riemannian manifold. I don’t even know what an “integral Riemannian manifold” is!

Put another way, I know how the curvature 2-form of a line bundle with connection can equal the 2-form that’s part of a Kähler structure - but I don’t know how the curvature 3-form of a gerbe with 2-connection can equal “the 3-form that’s part of a Riemannian structure”. There is no 3-form as part of a Riemannian structure!

Well, I suppose if my Riemannian manifold happened to be oriented and 3-dimensional, its volume form would be a 3-form. Then, if the deRham cohomology class of this 3-form were integral, I could say I had an integral Riemannian 3-manifold - and on this 3-manifold there’d be a gerbe with 2-connection whose curvature 3-form equalled the volume 3-form!

But it’s annoying to be limited to 3-manifolds here.

Posted by: John Baez on October 20, 2006 8:01 AM | Permalink | Reply to this

Re: A Course on Classical Mechanics

So, if XX is a Riemannian manifold, the free loop space LXL X becomes a Riemannian manifold as well.

And the cool thing is that the Nambu-Goto action of the string is precisely such that that it describes “geodesic motion up to loop reparameterization” on LXL X with respect to precisely this induced metric.

In this case, the space of smooth maps C (Σ,X)=LXC^\infty(\Sigma,X) = L X is not just a Riemannian manifold. It’s a Kähler manifold!

And it’s that Kähler structure of LXL X which is responsible for the “chirality” of the string, i.e. the splitting into “left- and right-moving” parts of the maps S 1XS^1 \to X.

Kähler manifolds are just what you want for geometric quantization.

This now leads to an interpretation of LXL X quite orthogonal to the one I just mentioned.

We can think of LXL X as the configuration space of the string. The Kähler structure then leads to chiral splitting.

Now, you want to interpret LXL X as the phase space of something.

[…] But the analogy is weird: […]

Maybe we should recall what Rovelli figured out about how to quantize with an integral 3-form.

He describes how to construct something like the phase space of a string by starting with an extended configuration space, taking the second exterior power of the cotangent bundle over it CC, and considering a 3-form ω\omega on that (to be thought of as coming from a gerbe).

Then the ordinary phase space of the string is recoved by taking loops in CC, pulling back ω\omega to S 1×LCS^1 \times L C and integrating over S 1S^1. The result is a 2-form which is supposed to the symplectic 2-form on the infinite-dimensional ordinary phase space of the string.

Posted by: urs on October 20, 2006 10:51 AM | Permalink | Reply to this

Re: A Course on Classical Mechanics

Yes, all that Rovelli stuff is more clearly sensible than the weird stuff I was dreaming about.

Heuristically, what sort of classical theory might you get if you thought of LXL X as a phase space? I’m guessing it’s one “chiral half” of the string theory you get by thinking of LXL X as a configuration space, which gives the cotangent bundle T *LXT^* L X as phase space.

After all, LXL X is “half as big” as T *LXT^* L X.

And, if we take X=X = \mathbb{R}, T *LXT^* L X is the phase space for the wave equation on a cylinder ×S 1\mathbb{R} \times S^1, while LXL X is the space of initial data for left-moving (or right-moving) solutions of the wave equation.

So, this should generalize to a bunch of other manifolds XX.

Posted by: John Baez on October 20, 2006 10:41 PM | Permalink | Reply to this

Re: A Course on Classical Mechanics

what sort of classical theory might you get if you thought of LXL X as a phase space?

I’m guessing it’s one “chiral half” of the string theory you get by thinking of LXL X as a configuration space

[…]

LXL X is the space of initial data for left-moving (or right-moving) solutions of the wave equation

Okay, let me see:

On T *LXT^* L X we locally have coordinates consisting of functions usually called

(1)P μ(σ) P_\mu(\sigma)

(momentum density along μ\partial_\mu at point σ\sigma on the string) and

(2)X μ(σ) X^{'\mu}(\sigma)

(σ\sigma-derivative along the string at point σ\sigma) and

(3)X cm μ X_{\mathrm{cm}}^\mu

(the “center-of-mass” coordinates).

These combine into the “chiral” coordinates consisting of

(4)X μ ±(σ):=P μ(σ)±Tg μν(X(σ))X ν(σ), X^\pm_\mu(\sigma) := P_\mu(\sigma) \pm T g_{\mu\nu}(X(\sigma))X'^\nu(\sigma) \,,

(where TT is the string tension).

Up to an issue with center-of-mass modes (which either leds to “level matching” for the closed string or to boundary conditions for the open string) this can presumeably be thought of as coordinatizing two copies of LXL X, as you say.

(I realize, though, that it’s been a while since I thought about these particular issues in detail.)

Posted by: urs on October 23, 2006 11:51 AM | Permalink | Reply to this

Re: A Course on Classical Mechanics

I just said it was annoying how we’re limited to 3-manifolds if we’re trying to get gerbes from Riemannian manifolds the way we get line bundles from Kähler manifolds.

But, I should emphasize that Brylinski - a smart fellow - makes precisely this limitation at a certain point in his book on gerbes and geometric quantization!

He gets some interesting stuff, namely a gerbey way of thinking about the dynamics of “vortex filaments” in 3d fluid flow. There might even be a nice way of geometrically quantizing these, if we could handle the infinite-dimensional Kähler manifold LXL X, where XX is our integral Riemannian 3-manifold.

Posted by: John Baez on October 20, 2006 8:08 AM | Permalink | Reply to this

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