Week 18: Symplectic and Contact Geometry

So, I finally hit paydirt and discovered symplectic and contact geometry a few weeks ago already, I guess. Symplectic manifolds are essentially time-independent ("autonomous") second-order ordinary differential equations on manifolds, and contact manifolds are essentially time-dependent ("non-autonomous") second-order ordinary differential equations on manifolds; both arise from Lagrangian mechanics in physics.

This, together with geometric control theory, is the holy grail of applications of differential manifolds for which I have been searching my whole life. The subject goes through Lagrangian mechanics, but from a differential topologist's point of view. I thought global analysis was the starting point for geometric control theory, but global analysis is essentially just first-order ordinary differential equations on manifolds, and second-order ODEs on manifolds (SODEs) are H***-and-gone from first-order ones; there's no Poincare-Hopf theorem, for a start.

I found a wonderful, free, paper/book online, Finsler-Lagrange Geometry by Bucataru and Miron, and I have been working through it the last for weeks, at this point, I guess. Like Transversal Mappings and Flows by Abraham and Robbin, it has some wonderful material on differential manifolds and their tangent and double-tangent bundles in the first chapter, but some of it is a bit of a math-out as I get to the final 3/4's of that chapter; I will definitely need to work through some examples of the transition functions from the change-of-coordinate-patch formulas - hopefully, it won't take me the 5 weeks it took to understand the canonical flip on the double-tangent bundle from Abraham and Robbin.

The material is just delightful. I finally understand the covariant derivative of a vector field along a curve (it can be defined with just a non-linear connection, but also surely with a Levi-Civita connection on a Riemannian manifold). I like the pedagogy of defining parallel transport along smooth (and, extended to piecewise-smooth) path and covariant derivative first, then defining the connection and adding things like symmetry (torsion-free), linearity, and homogeneity later; you can really see that you are differentiating a vector field along a curve by parallel transporting the nearby vectors back to the tangent space at a given point p and then doing the Newton quotient in that tangent space to get the derivative vector. This makes the notion of the second derivative of the intrinsic velocity vector field of a curve intuitive and makes you see that the curves with the covariant derivative of the velocity vector field being zero are the natural candidates for geodesics.

I still have to clean up the math-out from the end of chapter 1, and I really need to understand vertical and horizontal vectors in the double-tangent bundle - The Geometry of Jet Bundles by Sanders is just wonderful for these things, because he states them for arbitrary vector bundles, so you can compare and contrast with other authors who only cover the tangent and double-tangent bundle.

It'll still be a few chapters before I get to actual Lagrangians and sympletic/contact forms, but I really feel like I've found my calling in life: I truly find this mathematics "beautiful", and that is for what my major professor asked me to search.

Week 15: The Canonical Flip

So, I've been trying to read through Transversal Mappings and Flows by Abraham and Robbin (Joel Robbin was my linear algebra professor at UW back when I was an undergrad, and he literally wrote the book on the subject: we used a manuscript that I still have to this day that eventually became a book Matrix Algebra Using MINImal MATlab. The manuscript had about a third of it devoted to pure linear algebra, over abstract finite dimensional real and complex vector spaces, not using matrices; the theorems were mainly restatements of theorems from the matrix algebra portion of the book, and referenced their matrix algebra counterparts, but this portion of the book was written like manifold theory using manifold that were all diffeomorphic to R^n, and it really set the tone for a lot of math I was to learn later) in my efforts to learn global analysis (I've had a really hard time finding a good introduction to the subject, and this one seems as good as any, maybe better than most). The book talks about the double tangent bundle and its relations to second order ordinary (autonomous) differential equations, and front and center is the "canonical flip" on the double tangent bundle.

The Wikipedia article on the double tangent bundle also mentions the canonical flip, but I couldn't make heads or tails of it. I have been trying to read through Robbin's presentation of it since, oh, roughly since I made my last blog post, in December 2015, but I just couldn't wrap my head around it.

Then, finally, it came to me yesterday as I was riding the elevator back up to my parents' condo after having pored over the section on the double tangent bundle in Abraham and Robbin at a coffeshop for the umpteen-millionth time. In the book The Geometry of Jet Bundles by D. L. Saunders (a book on partial differential equations on manifolds, yet another book on global analysis I'm nominally trying to work through in my efforts to transition from manifold topology [MSC 57] to global analysis [MSC 58] and beyond that to geometric control theory [no MSC code yet, AFAIK]), he goes over in detail that, for an arbitrary vector bundle \pi: E -> M, in considering the tangent bundle to E, D\pi: TE -> TM, we have that for the horizontal vector bundle - that is, if (p, v1, v2, v3) is an arbitrary point of TE, the horizontal vector bundle is {(p, v2) | (p, v1, v2, v3) \in TE} - is canonically vector bundle isomorphic to TM - and, I understood the canonical flip! It just interchanges TM = {(p, v1) | (p, v1, v2, v3) \in TE} \subseteq TE with the horizontal vector bundle {(p, v2) | (p, v1, v2, v3) \in TE}.

So, persistence and hard work does sometimes pay off.

Now, Abraham and Robbin go on to define a second-order (autonomous) ordinary differential equations as dual sections of \tau(TM) and D\tau(M), and they, just before that, show that D: \Gamma(\tau(M) -> \Gamma(D\tau(M) and \omega \circ D: \Gamma(\tau(M) -> \tau(TM) are linear (over real vector bundles), so if \xi: M -> TM is an first-order (autonomous) ordinary differential equation on M, then (say) (1/2)D\xi + (1/2)(\omega \circ D\xi) is a second-order (autonomous) ordinary differential equation on M - nifty stuff.