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 (higher-order) numerical methods are recovered and canonically constructed by using different discretization maps, as well as some operations with Lagrangian submanifolds.

Lagrangian reduction of forced discrete mechanical systems

“…mapping the zero section onto the diagonal. There are several specializations of this basic idea, as the discretization maps of [1].…”

Feedback Integrators for Mechanical Systems with Holonomic Constraints

“…Another approach starts from the Hamiltonian or variational nature of mechanical systems to come up with discretizations that preserve the symplectic structure and the constraint manifold [5,[8][9][10]. Such discretizations, the RATTLE algorithm [11] and its generalizations [12][13][14] chiefly among them, typically exhibit superior long-term integration properties compared with standard, nonsymplectic integration algorithms, and have found wide application in the numerical integration of mechanical and control systems [15][16][17]. This approach has also been extended to the case of classical field theories with constraints [18,19], or to systems with dissipation [20].…”

Feedback Linearization of Implicit Discrete Systems

“…Before stating the main result, we provide a rapid refresher on the retraction and discretization maps. For more information on these maps one may look into [28] and references therein.…”

“…Definition 3.2 (Discretization Maps [28]): Let O ⊂ T M be an open neighborhood of the zero section of the tangent bundle…”