Variational methods, multisymplectic geometry and continuum mechanics

A Frame Bundle Generalization of Multisymplectic Momentum Mappings

This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum observables are vector-valued momentum mappings on the vertically adapted frame bundle generated from automorphisms of $Y$. The generalized symplectic geometry on $L_VY$ is a covering theory for multisymplectic geometry on the multiphase space $Z$, and it follows that the field mome…

多辛几何中的Floer截面

Floer sections in multisymplectic geometry

本文在多辛几何中引入了新的框架，将辛几何中的椭圆方法推广，并证明了伪Fueter曲线的存在。

多辛几何、变分积分器与非线性偏微分方程

Multisymplectic geometry, variational integrators, and nonlinear PDEs

本文基于多辛几何与变分原理，提出统一框架以构造保持辛性与诺特对称性的连续及离散力学/场论模型，并给出非线性正弦-戈登方程的数值验证。

用闭微分形式推广哈密顿力学

Generalizing Hamiltonian Mechanics with Closed Differential Forms

本研究针对经典哈密顿力学隐藏其他不变量、Nambu力学缺乏合适几何框架的问题，通过2次以上微分形式的可逆性建立广义哈密顿力学与多辛几何的新对应，重新解释核心定理并提出两个关键定理，推进力学几何框架的推广。

Phase Flows and Vector Hamiltonians

We present a generalization of the Nambu mechanics on the base of Liouville's theorem. We prove that the Poisson structure of an n-dimensional multisymplectic phase space is induced by (n-1)-Hamiltonian k-vector field seach of which requires introduction of k-Hamiltonians.

Manifolds of mappings for continuum mechanics

This is an overview article. After an introduction to convenient calculus in infinite dimensions, the foundational material for manifolds of mappings is presented. The central character is the smooth convenient manifold $C^{\infty}(M,N)$ of all smooth mappings from a finite dimensional Whitney manifold germ $M$ into a smooth manifold $N$. A Whitney manifold germ is a smooth (in the interior) manifold with a very general boundary, but still admitting a continuous Whitney extension operator. This notion is developed here for the needs of geometric continuum mechanics.