February 20 - May 8, 2014
In smooth dynamics of Lie groups, it's a consequence of the Frobenius Theorem that locally free actions have a foliated orbit structure. This allows for the formation of local transversals which reduces the classification of the orbits down to a problem in countable equivalence relations. For actions that are not locally free, this fails, but it's natural to ask if one can work around this difficulty by finding a related action that is locally free, and, with luck, even free. Passage to higher order frame bundles (and various associated bundles) is one technique for implementing this work-around. We will discuss situations where freeness can be found in frame bundles, and situations where it cannot.
In this talk, I will describe some of the uses of the invariant
variational bicomplex structure in studying the evolution of
differential invariants such as curvature, torsion, mean curvature, etc.
under a group invariant flow. This talk will also discuss
integrability of the resulting equations describing these evolutions,
and in the case of curves in Euclidean 3-space, the connection to the
non-linear Schrödinger equation via the Hasimoto transform.
In this talk I will use the method of moving frames to construct a group invariant version of the variational bicomplex. Explicit formulae for the group invariant versions of the Euler-Lagrange equations associated to an invariant variational problem arise in this context. If time permits, I will give further applications to invariant curve and surface flows in low dimensions. The talk is based on papers by Kogan and Olver and leads into the problems considered in my dissertation.
This week seminar will be devoted to the inverse problem of calculus of variations: 1) given a system of equations, decide whether or not it comes from a variational principle; 2) if the answer is positive, find a corresponding Lagrangian. I will present Vainberg, Volterra solution to this problem, as well as a solution in terms of the variational bicomplex. I will not prove the results but illustrate them by examples.
Pattern forming PDE often rely on a Lyapunov functional for uniqueness and existence, in turn the existence of such functionals often rely on infinitely extended domains. Exotic boundary conditions break this variational structure. I will present an overview of the Cahn-Hilliard equation and Swift-Hohenberg equation as examples of such pattern forming differential equations. For small bifurcation parameter values in the Swift-Hohenberg equation it is well known that there exists a family of solutions parameterized by the wavenumber. For the Swift-Hohenberg equation I will show a Numerical Homotopy of this family of solutions from Neumann to Transparent boundary conditions on a simulated half line which exhibit a transition from phase selection to wavenumber selection.
I will present the basics of jet space, contact forms, and the variational bicomplex. Applications to the calculus of variations, the inverse problem, conservation laws, and symmetries, possibly getting to Noether's Theorem(s), will be discussed. No prior knowledge beyond multivariable calculus and a little familiarity with differential forms will be assumed.
I will present the basics of jet space, contact forms, and the variational bicomplex. Applications to the calculus of variations, the inverse problem, conservation laws, and symmetries, possibly getting to Noether's Theorem(s), will be discussed. No prior knowledge beyond multivariable calculus and a little familiarity with differential forms will be assumed
I will review a physically-oriented approach to constraints and
symmetries in the variational bicomplex developed by Barnich, Fulp,
Henneaux, Lada, McCloud, and Stasheff.