Alex wrote:

Arnold talked a lot about contact geometry and its connections with classical mechanics, thermodynamics, optics and Hamilton–Jacobi, so it’s probably in there somewhere.

Maybe, but I don’t know where. I think the basic ideas are pretty simple, and a lot of what Arnold writes is more complicated.

The only problem is, (I’ve read the descriptions in all of these places and still) I don’t understand what contact geometry is supposed to be.

Symplectic geometry is easiest to understand if you think of it as a generalization of the geometry that any cotangent bundle has—that is, the geometry of position-momentum pairs.

Similarly, contact geometry is easiest to understand if you think of it as a generalization of the geometry that the trivial circle bundle over the cotangent bundle has—that is, the geometry of position-momentum-phase triples. This space is what we *really* should call “phase space”! Yes, phases are important in classical mechanics too.

I see that I explained this more carefully here 3 years ago.

Why would anyone want such a thing—in particular, why do we specifically care about “metric” symplectic geometry in the even-dimensional case but “conformal” contact geometry in the odd-dimensional case?

Hmm, maybe because *another* motivating example of a contact manifold is the projectivized cotangent bundle. The symplectic structure on the cotangent bundle puts a “nondegenerate 2-form up to an undetermined scale factor” on this.

So, starting from the cotangent bundle there’s a nice way to get a manifold of dimension one more (the circle bundle over it) and and a manifold of dimension one less (its projectivization), both of which are contact manifolds. There’s more about the seccond constructon here, but apparently not the first:

• Wikipedia, Contact geometry: relation with symplectic structures.

They also mention a third: a surface of constant energy in a cotangent bundle can sometimes be made into a contact manifold.

]]>Arnold talked a lot about contact geometry and its connections with classical mechanics, thermodynamics, optics and Hamilton–Jacobi, so it’s probably in there somewhere.

If I understood contact manifolds, I’d probably find the relevant parts in *Mathematical methods of classical mechanics* appendix 4, “Contact structures”. Other places he talks about contact manifolds are *Geometric methods in the theory of ordinary differential equations* §8, “The non-linear first-order partial differential equation”; *Lectures on partial differential equations* lecture 4, “Huygens’ principle in the theory of wave propagation”; and *Topological invariants of plane curves and caustics* lecture 2, “Symplectic and contact topology of caustics and wave fronts”.

The only problem is, (I’ve read the descriptions in all of these places and still) I don’t understand what contact geometry is supposed to be.

If I look for something “flat” (linear-algebraic) that contact geometry is supposed to be the “curved” (differential-geometric) version of, the best I can do is that it is… a conformal (up-to-scalar-factor) geometry of maximal-rank antisymmetric bilinear forms in odd dimension? Why would anyone want such a thing—in particular, why do we specifically care about “metric” symplectic geometry in the even-dimensional case but “conformal” contact geometry in the odd-dimensional case?

]]>I have MathJax set up on my WordPress site (which I host on my own website, rather than on WordPress.com). Here’s an example post with formulae in it.

I don’t have experience with blogs hosted on WordPress.com, but you may be able to add MathJax support by editing the theme’s header file.

]]>The current draft is here. I’m moving very slowly on it, e.g. right now not at all.

]]>You’re writing a book on classical mechanics? Awesome! Keep us updated.

]]>I spent several years trying to support the same technology that you see at the nCafé on WordPress. Eventually, I abandoned the effort because the WordPress codebase was too unbearably crappy to make the cost/benefit worthwhile.

MathJax (which, of course, we use to support browsers without native MathML support) may not be great, but it’s *way* better than crappy little GIF images.

So let me present the same construction again, more cleanly.

]]>Yes. In this article (okay, post) I explained the concept of a Chern connection:

]]>2) We can demand that the connection be compatible with the holomorphic structure of Let me spell this out a little. Since is a holomorphic line bundle there’s a operator that we can apply to any section of and get an -valued 1-form. This is an -valued (0,1)-form, meaning that written out in holomorphic complex coordinates it has terms but no terms. On the other had, we can use the connection to take the covariant derivative of which is a -valued 1-form and take its (0,1) part, denoted Then, we can demand that these agree:

3) We can demand that the connection be hermitian. This means that the directional derivative of the inner product of two sections of can be computed using a kind of product rule where we differentiate each section using Namely:

Any holomorphic hermitian line bundle has a unique connection obeying 2) and 3). This is called the

Chern connectionand its construction is actually given here:• Wikipedia, Hermitian metrics on a holomorphic vector bundle.