A variational principle for volume-preserving dynamics

Helicity is the only integral invariant of volume-preserving transformations

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mathcal I$ is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.

Volume preserving multidimensional integrable systems and Nambu-Poisson Geometry

In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu-Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. Recently Takasaki-Takebe provided the twistor construction of dispersionless KP and dToda type equations by using the Gindikin's pencil of two forms. In this paper we generalize this twistor construction to our systems.

General Volume-Preserving Mechanical Systems

In this letter, we present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, ω). It is shown that every volume-preserving flow has some 2-forms acting the role of the Hamiltonian functions in the Hamiltonian mechanics and the ordinary Hamilton equations are included as a special case with a 2-form \frac{1}{n-1} H ωwhere H is the corresponding Hamiltonian.

关于无穷体积中欧几里得量子场的变分方法

On the variational method for Euclidean quantum fields in infinite volume

本文研究无穷体积下欧几里得量子场的变分方法，证明二维$φ^4_2$理论的紧性与偏差原理，并用弱形式正反向随机微分方程刻画无穷体积测度，还在完整$L^1$区域证明了$exp(βφ)_2$模型无穷体积极限的唯一性。

变分场理论的对称保持离散化

Symmetry-preserving discretization of variational field theories

本文发展了定义在抽象细胞复形上的离散场变分理论，通过离散拉格朗日密度的变分原理推导离散公式，并研究了离散场方程的对称性和守恒律，最后通过柯西问题的变分积分器和柯萨特杆动力学的离散化示例展示理论。

Variational Framework for Structure-Preserving Electromagnetic Particle-In-Cell Methods

In this article we apply a discrete action principle for the Vlasov--Maxwell equations in a structure-preserving particle-field discretization framework. In this framework the finite-dimensional electromagnetic potentials and fields are represented in a discrete de Rham sequence involving general finite element spaces, and the particle-field coupling is represented by a set of projection operators that commute with the differential operators. With a minimal number of assumptions which allow for a variety of finite elements and shape functions for the particles, we show that the resulting varia…