So, in my undergraduate biology text, there was a chapter entitled, "Biology, Having Found Its Holy Grail, Drinks Deeply From It" (at least, that's how I remember the chapter title), and that sums up my last week. I found a book at the library, Foundations of Mechanics by Abraham and Marsden (the same Abraham as in Transversal Mappings and Flows by Abraham and Robbin), and it has a wonderful exposition of symplectic manifolds as it pertains to Lagrangian and Hamiltonian mechanics. It turns out, associated to the cotangent bundle, T*(Q), of a configuration space (or smooth manifold, as we differential topologists like to call them) Q, there is a canonical 1-form -θ, and its exterior derivative, ω = -dθ, which is the canonical symplectic form associated to T*(Q). All the hullabaloo associated to symplectic and contact geometry stems from this fact.
So, it took me all of Monday (05/02/2016) after class to wrap my brain around the definition of the canonical 1-form and symplectic form: evidently, if (q, u*, v, w) denotes the general point of TT*(Q), then θ = u*(v) (using the canonical flip on TT(M), around which took me 5 months to wrap my brain - nice to see some of that theory pay off), and ω is then -dθ. A technically precise, but difficult to understand, definition of the canonical 1-form is that it is (u*)o(Dτ*Q)(q,u*), where τ*Q is the cotangent bundle projection map. To say the same thing slightly differently, as it is tradition to use q to represent an arbitrary point of Q ("generalized coordinates") and α to represent an arbitrary (co)tangent vector at q and using a c-patch φ with φ(q, α) = (q1(q, α), ..., q1(q, α), p1(q, α), ...., pn(q, α)), we have θ0 = p1dq1 + ... + pndqn so -dθ0 =-(dp1∧dq1 + ... + dpn∧dqn) = dq1∧dp1 + ... + dqn∧dpn = ω - neat.
So, I also discovered this great video series on YouTube on symplectic and contact geometry:
The first video has a nice overview of symplectic geometry, most of which I already knew, but the second video takes a dog-leg into things more "theory of symplectic manifold"-oriented and less "building up to geometric control theory"-oriented, so I'm not sure how much more of the series I'm going to watch.
So, that's it for this week. I hope to start on Geometric Control of Mechanical Systems by Bullo and Lewis soon, and then onto trying to try to publish something in the field.
