Variational integrators for Euler--Lagrange equations and Hamilton's equations are a class of structure-preserving numerical methods that respect the conservative properties of such systems. Lie group variational integrators are a particular class of these integrators that apply to systems which evolve over the tangent bundle and cotangent bundle of Lie groups. Traditionally, these are constructed from a variational principle which assumes fixed position endpoints. In this paper, we instead construct Lie group variational integrators with a novel Type II variational principle on the cotangent bundle of a Lie group which allows for Type II boundary conditions, i.e., fixed initial position and final momenta; these boundary conditions are particularly important for adjoint sensitivity analysis, which is the motivating application in our paper. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. By developing the continuous and discrete Type II variational principles over Lie groups, we construct a structure-preserving Lie group variational integrator that is both symplectic and momentum-preserving. Subsequently, we introduce adjoint systems on Lie groups, and show how these adjoint systems can be used to perform geometric adjoint sensitivity analysis for optimization problems on Lie groups. Finally, we conclude with two numerical examples to show how adjoint sensitivity analysis can be used to solve initial-value optimization problems and optimal control problems on Lie groups.

翻译：欧拉-拉格朗日方程和哈密顿方程的变分积分器是一类保持系统保守性质的结构保持数值方法。李群变分积分器是这类积分器中特殊的一类，适用于在李群的切丛和余切丛上演化的系统。传统上，这些积分器基于假设位置端点固定的变分原理构建。本文另辟蹊径，利用李群余切丛上新型的第二类变分原理构建李群变分积分器，该原理允许第二类边界条件，即固定初始位置和最终动量。此类边界条件对于伴随敏感性分析尤为关键，这也正是本文的核心应用动机。一般而言，第二类变分原理仅在向量空间上全局定义，或在一般流形上局部定义。然而，通过左平移变换，我们能够在李群余切丛上全局定义该变分原理。通过发展李群上的连续和离散第二类变分原理，我们构建了兼具辛性和动量保持性的结构保持李群变分积分器。随后，我们引入李群上的伴随系统，并展示如何利用这些伴随系统对李群上的优化问题进行几何伴随敏感性分析。最后，通过两个数值算例，我们演示了伴随敏感性分析如何用于求解李群上的初值优化问题和最优控制问题。