A symplectic realization of the Volterra lattice

Mass, Impetus and Force by Symplectic Realizations of the Static Group

We show by symplectic realizations of the one dimensional Static group $G$ that the maximal $G$-elementary system is a a massive particle under an invariant force $f$ participating in the linear momentum and an invariant impetus $I$ participating in the change of position. Moreover the system is characterized by four physical quantities : a mass $m$, a boost $u=\frac{I}{m}$, an acceleration $a=\frac{f}{m}$ and an internal energy $U$. The hamiltonian of the system is a sum of the potential energy $V=-fq$ and a contribution $pu$ of the impetus.

Poisson电动力学中的辛实现与李群胚

Symplectic realizations and Lie groupoids in Poisson Electrodynamics

本文以辛实现、李代数胚与辛群胚重新表述Poisson电动力学的规范势及规范变换，并由正交叶状结构导出协变电磁二形式及其时空—动量对偶。

Generalized Volterra lattices: binary Darboux transformations and self-consistent sources

We study two families of (matrix versions of) generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete variables, which is a realization of a general integrable equation in bidifferential calculus. This allows to derive a binary Darboux transformation and also self-consistent source extensions via general results of bidifferential calculus. Exact solutions are constructed from the simplest seed solutions.

Integrable Hamiltonian systems on the symplectic realizations of $\textbf{e}(3)^*$

The phase space of a gyrostat with a fixed point and a heavy top is the Lie-Poisson space $\textbf{e}(3)^*\cong \mathbb{R}^3\times \mathbb{R}^3$ dual to the Lie algebra $\textbf{e}(3)$ of Euclidean group $E(3)$. One has three naturally distinguished Poisson submanifolds of $\textbf{e}(3)^*$: (i) the dense open submanifold $\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset \textbf{e}(3)^*$ which consists of all $4$-dimensional symplectic leaves ($\vecΓ^2>0$); (ii) the $5$-dimensional Poisson submanifold of $\mathbb{R}^3\times \dot{\mathbb{R}}^3$ defined by $\vec{J}\cdot \vecΓ = μ||\vecΓ||$; (iii) th…

$κ$-Minkowski star product in any dimension from symplectic realization

We derive an explicit expression for the star product reproducing the $κ$-Minkowski Lie algebra in any dimension $n$. The result is obtained by suitably reducing the Wick-Voros star product defined on $\mathbb{C}^{d}_θ$ with $n=d+1$. It is thus shown that the new star product can be obtained from a Jordanian twist.

Symplectic realizations of holomorphic Poisson manifolds

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, π)$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^*X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic s…

广义Snyder-Poisson代数的辛实现

Symplectic realization of generalized Snyder-Poisson algebra

本文在经典极限下系统研究Snyder时空及其推广的Poisson规范理论，构造其辛实现，提出用正则相空间变量表示Snyder代数的通用方法，并分析自由落体运动与空间几何。

Recurrence relations for symplectic realization of (quasi)-Poisson structures

It is known that any Poisson manifold can be embedded into a bigger space which admites a description in terms of the canonical Poisson structure, i.e., Darboux coordinates. Such a procedure is known as a symplectic realization and has a number of important applications like quantization of the original Poisson manifold. In the present paper we extend the above idea to the case of quasi-Poisson structures which should not necessarily satisfy the Jacobi identity. For any given quasi-Poisson structure $Θ$ we provide a recursive procedure of the construction of a symplectic manifold, as well as t…

On the symmetries of a Rikitake type system

A symplectic realization and some symmetries of a Rikitake type system are presented.

Symplectic realization of two interacting spin-two fields in three dimensions

We constructed a symplectic realization of the dynamic structure of two interacting spin-two fields in three dimensions. A significant simplification refers to the treatment of constraints: instead of performing a Hamiltonian analysis $\grave{a}\, la$ Dirac, we worked out a method that only uses properties of the pre-symplectic two-form matrix and its corresponding zero-modes to investigate the nature of constraints and the gauge structure of the theory. For instance, we demonstrate that the contraction of the zero-modes with the potential gradient, yields explicit expressions for the whole se…

The ubiquity of the symplectic hamiltonian equations in mechanics

In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry characterizing the different systems. In particular, we obtain that the class of the so-called algebroids covers a great variety of mechanical systems. Finally, as the main result, a hamiltonian symplectic realization of systems defined on algebroids is obtained.

四维可积映射与修正Volterra格

Integrable maps in 4D and modified Volterra lattices

本文指出Gubbiotti等人分类中的两个四维可积映射分别对应两种修正Volterra格的亏格二解，并通过Miura型变换与$g=2$的Volterra映射${\cal V}_2$相关联。

二维典范系统的自伴随实现及其应用

Self-adjoint realizations of 2d-dimensional canonical systems and applications

本文研究二维典范系统的自伴随实现及其在偏微分方程谱问题中的应用，包括行波稳定性分析和非线性薛定谔方程的线性化。

Invariant Poisson Realizations and the Averaging of Dirac Structures

We describe an averaging procedure on a Dirac manifold, with respect to a class of compatible actions of a compact Lie group. Some averaging theorems on the existence of invariant realizations of Poisson structures around (singular) symplectic leaves are derived. We show that the construction of coupling Dirac structures (invariant with respect to locally Hamiltonian group actions) on a Poisson foliation is related with a special class of exact gauge transformations.

Quantization of Magnetic Poisson Structures

We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non-geometric fluxes. We survey approaches based on deformation quantization of twisted Poisson structures, symplectic realization of almost symplectic structures, and geometric quantization using 2-Hilbert spaces of sections of suitable bundle gerbes. We compare and contrast these perspectives, describing their advantages and shortcomings in each case, and mention many ope…

Jacobi Hamiltonian Integrators: 构造与应用

Jacobi Hamiltonian Integrators: construction and applications

本文提出了一种构造保持结构的JacobiHamilton积分器的系统框架，并展示了其在接触Hamilton系统和经典模型中的应用及优势。