约束高阶拉格朗日-庞加莱方程的变分离散化

关于 Euler、Lagrange 和 Kovalevskaya 刚体运动可积情形的张量不变量

On tensor invariants for integrable cases of Euler, Lagrange and Kovalevskaya rigid body motion

本文研究了 Euler、Lagrange 和 Kovalevskaya 刚体运动可积情形的全局张量不变量，通过将含三次项的张量场代入不变性方程并用计算机代数系统求解，基于 Poincaré-Cartan 理论探讨了利用这些不变几何结构研究动力学的问题。

On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems

We study the secular evolution of several exoplanetary systems by extending the Laplace-Lagrange theory to order two in the masses. Using an expansion of the Hamiltonian in the Poincaré canonical variables, we determine the fundamental frequencies of the motion and compute analytically the long-term evolution of the keplerian elements. Our study clearly shows that, for systems close to a mean-motion resonance, the second order approximation describes their secular evolution more accurately than the usually adopted first order one. Moreover, this approach takes into account the influence of the…

Hamel方程及约束与浮动多体系统和空间系统的几何力学

Hamel's Equations and Geometric Mechanics of Constrained and Floating Multibody and Space Systems

本文通过Hamel原始形式体系推导约化Euler-Lagrange方程与联络曲率，利用锁定速度实现内蕴分解与惯性解耦，填补了Hamel表述与几何方法在局部坐标下联系的空白。

关于Suslov问题离散化的某些方面

On some aspects of the discretization of the Suslov problem

本文探讨了在SO(3)上Euler-Poincaré-Suslov方程的离散化，分析了不约简与约简设置下离散重构方程由Cayley重映射给出时的一致性阶关系，并给出了保证变分积分器保持约束分布的条件，同时通过图表展示了不同离散化方法的性能。

含碰撞力学系统的非完整约化

Nonholonomic reduction for mechanical systems with collisions

本文研究主丛上含碰撞非完整系统的非光滑变分问题，推广变分原理得到隐式拉格朗日-达朗贝尔-庞特里亚金方程，并利用联络实现对称性约化。

切丛上的狄拉克结构：变分原理、约束动力学和对称约化的几何框架

Dirac structures on tangent bundles: a geometric framework for variational principles, constrained dynamics, and symmetry reduction

提出切丛上的拉格朗日-狄拉克结构，统一描述非完整、退化拉格朗日和对称系统，并建立拉格朗日-达朗贝尔-狄拉克变分原理及李群对称约化理论。

Applications of Poisson Geometry to Physical Problems

We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincaré equations. In this case, the invariant Lagrangian is defined on the Lie algebra of the group and its Euler-Poincaré equations are defined on the dual Lie algebra, where dual is defined by the operation of taking variational derivative. On the Hamiltonian side, the Euler-Poincaré equations are Lie-Poisson and they possess accompanying momentum maps, which enco…

Variational discretization of thermodynamical simple systems on Lie groups

This paper presents the continuous and discrete variational formulations of simple thermodynamical systems whose configuration space is a (finite dimensional) Lie group. We follow the variational approach to nonequilibrium thermodynamics developed in \cite{GBYo2017a,GBYo2017b}, as well as its discrete counterpart whose foundations have been laid in \cite{GBYo2017c}. In a first part, starting from this variational formalism on the Lie group, we perform an Euler-Poincaré reduction in order to obtain the reduced evolution equations of the system on the Lie algebra of the configuration space. We o…

Stochastic fiber dynamics in a spatially semi-discrete setting

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit represe…