Retraction maps: a seed of geometric integrators

TL;DR

This paper introduces the concept of discretization maps, extending retraction maps, to develop geometric integrators that preserve symplectic structure and improve numerical methods for differential equations.

Contribution

It rigorously defines discretization maps and their lifts, enabling the construction of higher-order geometric integrators using tangent and cotangent bundle geometry.

Findings

Discretization maps generalize retraction maps for geometric integration.

Cotangent lifts of discretization maps are symplectomorphisms.

The framework recovers and constructs various high-order numerical methods.

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.