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Retraction Maps: A Seed of Geometric Integrators
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2022-06-08 , DOI: 10.1007/s10208-022-09571-x
María Barbero-Liñán, David Martín de Diego

The classical notion of retraction map used to approximate geodesics is extended and rigorously defined to become a powerful tool to construct geometric integrators and it is called discretization map. Using the geometry of the tangent and cotangent bundles, we are able to tangently and cotangent lift such a map so that these lifts inherit the same properties as the original one and they continue to be discretization maps. In particular, the cotangent lift of a discretization map is a natural symplectomorphism, what plays a key role for constructing geometric integrators and symplectic methods. As a result, a wide range of (higher-order) numerical methods are recovered and canonically constructed by using different discretization maps, as well as some operations with Lagrangian submanifolds.



中文翻译:

回缩图:几何积分器的种子

用于近似测地线的缩回图的经典概念经过扩展和严格定义,成为构造几何积分器的有力工具,称为离散化图。使用切线束和余切束的几何形状,我们能够切线和余切提升这样的地图,以便这些提升继承与原始提升相同的属性,并且它们继续是离散化地图。特别是离散化映射的余切提升是一种自然的辛同胚,它对构造几何积分器和辛方法起着关键作用。结果,通过使用不同的离散化映射以及拉格朗日子流形的一些操作,可以恢复和规范地构建各种(高阶)数值方法。

更新日期:2022-06-09
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