Optimizing Formal Series Expansions for Symplectic Maps
Student: Brandon Gonzales
Dates of Involvement: 2007
Faculty Advisor: Dr. James Howard
Collaborator: Dr. James Meiss
Symplectic Maps
Many problems in science and mathematics have Hamiltonian structure. The solution set of all possible configurations of the Hamiltonian can be modeled on a manifold, which is an abstract space where every point has a neighborhood which resembles Euclidian space. A symplectic structure on such a manifold is defined as a closed nondegenerate differential 2-form ω = dp ∧ dq which is an antisymmetric bilinear form which acts on a pair of tangent vectors. A diffeo- morphism on a 2n−dimensional manifold with coordinates (p, q) is symplectic if it preserves the symplectic form [6]. An example is a one degree-of-freedom system where symplecticness is simply the preservation of oriented area [8]. Specifically, we will consider splittable Hamiltonians of the form H = T (p) + U (q), where T is the kinetic energy and U the potential energy of a system.
Flows and Maps
We are interested in flows that model a given map. Symplectic maps are often derived from Hamiltonian flows, and conversely symplectic maps have an associated Hamiltonian flow. Exploiting this correspondence, the numerical value of the associated Hamiltonian is an invariant. If a map is symplectic its associated generating function can be found. Formally expanding this generating function in powers of h will yield invariants in the neighborhood of a resonance. This series is in general divergent [3], but if the map is splittable, the Baker-Campbell- Hausdorff (BCH) formula and other Lie Methods can be used to generate terms to “correct” the Hamiltonian to better approximate the map. The approximate invariant Hamiltonian flow will converge to the form of the map as the order of the expansion is increased up to an optimal order of convergence.
Problem Statement and Results
There are many physical systems which can be described by symplectic maps. Systems which have localized forces in time or space, such as circular particle accelerators, storage rings, and Birkhoff's ideal billiard can be modeled by symplectic maps [6]. Well known maps such as the Fermi Map, Standard Map, Nontwist Maps, and Froéschle Map are all symplectic. As such, finding an optimal number of terms for each of these maps' Hamiltonian will yield even better approximations and be useful in further study of these systems.
A method for expanding the BCH formula symbolically and numerically was developed using the Mathematica application. The intent of this investigation was to derive an optimal number of terms for a given symplectic map. Some previous studies investigated the convergence of the Birkhoff-Gustavson normal form up to 50th order and it was assumed initially that the BCH series would be similarly simple to evaluate. However this was not the case. Finding a suitable form for the BCH formula was nontrivial as the series had to be expanded using matrix methods and converted to non-commutative form after which the Poisson bracket operation had to be evaluated recursively in the form of nested commutators. After the algorithm was constructed, no computer was able to generate any terms higher than 17th order without running out of memory (4GB Ram, multicore processors), and literature search revealed only a handful of published results which compute orders any higher than 10. This led to the conclusion that a computational limit is likely to be encountered before an asymptotic divergence with respect to the BCH formula occurs.
A numerical method to compute the divergence using standard deviations was developed and the size of the series for the first 13 orders was calculated. It was observed that just as the number of terms grew exponentially, the deviation from the mean of the standard map decreased exponentially near an elliptic fixed point. To display the results of the BCH approximation, an interactive program was designed to aid in displaying the approximation together with the map as the order of the expansion was increased using a graphical slider control.
Figure 1: Animated figure showing convergence of the asymptotic series to the map points.
Figure 2 Asymptotic expansion, Hamiltonian System investigated, and convergence plot.
About Brandon:
Brandon is an undergraduate double majoring in Applied Mathematics and Civil Engineering at The University of Colorado at Boulder. He is also a research assistant at the Laboratory for Atmospheric and Space Physics (LASP) working under Dr. James Howard. His research assignments at LASP include particle dynamics and simulation model enhancement. While at LASP he has modeled radiation pressure effects on water/ice particles and has performed analysis and numerical integration algorithm upgrades and assisted with publication duties. He is originally from Ouray, Colorado and plans to obtain his PhD soon in Applied Math.
