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Variational integrators for Euler--Lagrange equations and Hamilton'sequations are a class of structure-preserving numerical methods that respectthe conservative properties of such systems. Lie group variational integratorsare a particular class of these integrators that apply to systems which evolveover the tangent bundle and cotangent bundle of Lie groups. Traditionally,these are constructed from a variational principle which assumes fixed positionendpoints. In this paper, we instead construct Lie group variationalintegrators with a novel Type II variational principle on the cotangent bundleof a Lie group which allows for Type II boundary conditions, i.e., fixedinitial position and final momenta; these boundary conditions are particularlyimportant for adjoint sensitivity analysis, which is the motivating applicationin our paper. In general, such Type II variational principles are only globallydefined on vector spaces or locally defined on general manifolds; however, byleft translation, we are able to define this variational principle globally oncotangent bundles of Lie groups. By developing the continuous and discrete TypeII variational principles over Lie groups, we construct a structure-preservingLie group variational integrator that is both symplectic andmomentum-preserving. Subsequently, we introduce adjoint systems on Lie groups,and show how these adjoint systems can be used to perform geometric adjointsensitivity analysis for optimization problems on Lie groups. Finally, weconclude with two numerical examples to show how adjoint sensitivity analysiscan be used to solve initial-value optimization problems and optimal controlproblems on Lie groups.

Type II Hamiltonian Lie Group Variational Integrators with Applications to Geometric Adjoint Sensitivity Analysis