Seminario de Mecánica Geométrica y Control
A geometric seed for mechanical numerical integrators
Ponente: María Barbero-Liñán (UPM)Fecha: viernes 11 de junio de 2021 - 15:30Lugar: Online - zoom.us/j/96739216821?pwd=UDhPYlNzM0M3dXFHUnJRSW92enl1dz09
Ordinary differential equations are often solved by numerical methods. It is well-known in the literature that geometric integrators have strong and desirable properties to get a better perfomance. Typically, geometric integrators for mechanical systems are obtained by discretizing a variational principle, that is, an action to be minimized. Here, we focus on discretizing the configuration manifold by extending the notion of retraction map. Such maps are used to approximate trajectories of differential equations, to optimize functions, to interpolate, etc. Given an initial point and velocity, a retraction map returns another point to construct the sequence of points solution to the problem under study. On Riemannian manifolds the exponential map is an example of retraction map. Here we define an extended retraction map that given a point and a velocity returns two different points on the configuration manifold. They could be both different from the given one. We show in this talk how extended retraction maps generate geometric integrators by exploiting the geometry of the tangent and cotangent bundles. As a result, we obtain symplectic integrators for mechanical systems in the Hamiltonian and Lagrangian framework. This is a joint work with David Martín de Diego.