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arXiv · 2106.00607

Retraction maps: a seed of geometric integrators

María Barbero Liñán·David Martín de Diego
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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.

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María Barbero Liñán, David Martín de Diego. 2021-06-01. Retraction maps: a seed of geometric integrators. https://doi.org/10.1007/s10208-022-09571-x

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