Multisymplectic Geometry and Geometric Numerical Integrator for Variational Problems

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14365
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Keio University
• Principal Investigator: Peng Linyu 慶應義塾大学, 理工学部(矢上), 准教授 (90725780)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: Multisymplectic geometry / Geometric integration / Variational calculus / Symmetry / Noether's theorem / Conservation law / Lagrangian

Outline of Final Research Achievements

In the current project, we have delved into the variational analysis of differential and discrete equations from various perspectives. For variational differential equations, we have developed a discrete counterpart of the variational bicomplex structure. This framework aids in understanding fundamental geometric and algebraic features, such as multisymplectic structure, symmetries, and conservation laws. For nonvariational problems, we have defined a modified formal Lagrangian formulation (MFLF), enabling the treatment of any differential equations. Specifically, this approach facilitates the systematic construction of conservation laws using Noether’s Theorem and variational integration even for nonvariational equations.

Free Research Field

応用数学

Academic Significance and Societal Importance of the Research Achievements

離散variational bicomplexは、差分方程式や数値方法の幾何及び代数的な研究するための基本的なツールになっている。対称性、保存則、マルティシンプレクティック構造、逆問題などはvariational bicomplexのコホモロジー群に関係しています。また、非変分問題のためのmodified formal Lagrangian formulationが導入されており、Noetherの定理から保存則を導出し、変分積分法の構築を容易にしています。これらの革新的な構造と理論は、物理現象の理解と効率的な数値積分法の開発に期待されます。