Retraction maps: a seed of geometric integrators
Abstract
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 (higer-order) numerical methods are recovered and canonically constructed by using different discretization maps, as well as some operations with Lagrangian submanifolds.
Cite
@article{arxiv.2106.00607,
title = {Retraction maps: a seed of geometric integrators},
author = {María Barbero Liñán and David Martín de Diego},
journal= {arXiv preprint arXiv:2106.00607},
year = {2022}
}
Comments
Paper accepted for publication in Found. Comput. Math. in Feb. 2022