Phantom: A topological method to analyze macro-system and its singular perturbation

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13532
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical physics/Fundamental condensed matter physics
• Research Institution: The University of Tokyo
• Principal Investigator: Yoshida Zensho 東京大学, 大学院新領域創成科学研究科, 教授 (80182765)
• Research Collaborator: MORRISON J. Philip テキサス大学オースチン校, 物理学科, 教授 ; TOKIEDA Tadashi スタンフォード大学, 数学科, 教授
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: トポロジー束縛 / 自己組織化 / 葉層構造 / ハミルトン力学系 / カシミール元 / ヘリシティー / 階層構造 / 可積分

Outline of Final Research Achievements

Topological constraints play a key role in the self-organizing processes that create structures in macro systems. Some topological constraints are represented by Casimir invariants (such as helicities), and then, the effective phase space is reduced to the Casimir leaves. However, a general constraint is not necessarily integrable, which precludes the existence of an appropriate Casimir invariant. We have formulated a systematic method to embed a Hamiltonian system in an extended phase space; we introduce “phantom fields” and extend the Poisson algebra. A phantom field defines a new Casimir invariant, a cross helicity. This hierarchical relation of degenerate Poisson manifolds enables us to see the interior of a singularity as a sub Poisson manifold. The theory can be applied to describe bifurcations and instabilities in a wide class of general Hamiltonian systems.

Free Research Field

非線形科学，プラズマ物理学