切丛与约束拉格朗日系统的几何离散模拟
Geometric discrete analogues of tangent bundles and constrained Lagrangian systems
摘要：
离散化变分原理，与离散化微分方程相反，导致力学的离散时间模拟，并系统地导致几何数值积分器。这种变分离散化的相空间通常是组态对的集合，类似地对应于切向量的起点和终点。我们开发替代离散类似物的切束，通过延长切向量的有限曲线段，一个曲线段为每个切向量。对灵活的，高阶数值积分，我们使用这些离散的切丛作为相空间的拉格朗日系统的变分原理的离散化，非完整力学系统的非线性约束的一般性。我们得到了一个独立的和透明的发展，其中规则性，运动方程，对称性和动量，结构保存，都有自然的表达。
英文摘要：
Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vectors. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.
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。
