A rigid body dynamics derived from a class of extended Gaudin models : an integrable discretization

An integrable discretization of the rational su(2) Gaudin model and related systems

The first part of the present paper is devoted to a systematic construction of continuous-time finite-dimensional integrable systems arising from the rational su(2) Gaudin model through certain contraction procedures. In the second part, we derive an explicit integrable Poisson map discretizing a particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the contraction procedures enable us to construct explicit integrable discretizations of the continuous systems derived in the first part of the paper.

From su(2) Gaudin Models to Integrable Tops

In the present paper we derive two well-known integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) performing an algebraic contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin models. The procedure preserves the linear r-matrix formulation of the ancestor models. We give the Lax representation of the resulting integrable systems in terms of su(2) Lax matrices with rational and elliptic dependencies on the spectral parameter. We finally give some results about the many-body extensions of the constructed systems.

Separation of variables and Bäcklund transformations for the symmetric Lagrange top

We construct the 1- and 2-point integrable maps (Bäcklund transformations) for the symmetric Lagrange top. We show that the Lagrange top has the same algebraic Poisson structure that belongs to the $sl(2)$ Gaudin magnet. The 2-point map leads to a real time-discretization of the continuous flow. Therefore, it provides an integrable numerical scheme for integrating the physical flow. We illustrate the construction by few pictures of the discrete flow calculated in MATLAB.

Hirota-Kimura Type Discretization of the Classical Nonholonomic Suslov Problem

We constructed Hirota-Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories asymptotically tend to a line of discrete analogies of so-called steady-state rotations. The last property completely corresponds to well-known property of the continuous Suslov case. The explicite formulae for solutions are given. In n-dimensional case we give discrete equations.

Bäcklund Transformations for the Kirchhoff Top

We construct Bäcklund transformations (BTs) for the Kirchhoff top by taking advantage of the common algebraic Poisson structure between this system and the $sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an exact time-discretization of the system, inasmuch as they preserve both its Poisson structure and its invariants. Moreover, in some special cases we are able to show that these maps can be explicitly integrated in terms of the initial conditions and of the "iteration time" $n$. Encouraged by these partial results we make the conjecture that the maps are interpolate…

连续体中的嵌入结构：线性模型与有限元离散化

Embedding structures in continua: linear models and finite element discretizations

本文提出一个通用框架，用于将刚体、梁、壳等嵌入结构构件与连续体在能量意义上耦合，并证明问题及有限元近似适定，通过数值示例展示其通用性和稳健性。