离散狄拉克力学的变分与几何结构
Variational and Geometric Structures of Discrete Dirac Mechanics
AI总结：
本文通过构造Tulczyjew三元组和诱导狄拉克结构的离散类比，发展了退化拉格朗日/哈密顿约束系统的离散狄拉克力学理论，并统一了离散拉格朗日与哈密顿力学，推广了辛和泊松积分器至狄拉克积分器。
摘要：
本文发展了离散狄拉克力学的理论基础，即带有约束的退化拉格朗日/哈密顿系统的离散力学。我们首先通过考虑辛映射及其相关生成函数的几何结构，构造了Tulczyjew三元组和诱导狄拉克结构的离散类比。我们证明该框架提供了推导离散拉格朗日-狄拉克和非完整哈密顿系统的方法。特别地，这产生了非完整拉格朗日和哈密顿积分器。我们还引入了离散拉格朗日-达朗贝尔-庞特里亚金和哈密顿-达朗贝尔变分原理，它们提供了同一组积分算法的另一种推导。本文在更一般的离散狄拉克力学背景下统一处理了离散拉格朗日和哈密顿力学，并将辛和泊松积分器推广到更广泛的狄拉克积分器类别。
英文摘要：
In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
