Retraction maps are used in many research fields such as approximation of trajectories of differential equations, optimization theory, interpolation theory etc. In this talk we will review the concept of retraction map in differentiable manifolds to generalize it to obtain an extended retraction map from the tangent bundle of the configuration manifold to two copies of this manifold. After suitably lifting the new retraction map to the cotangent bundle, the typical phase space for Hamiltonian mechanical systems, we will be able to define geometric integrators for optimal control problems. This is a joint work with David Martín de Diego.

It is well-known that kinetic nonholonomic dynamics is neither Hamiltonian nor variational. However, in this talk, after introducing a geometrical description of nonholonomic dynamics, I will present a result which is a little surprising. Namely, for a kinetic nonholonomic system and a given point \(q\) of the configuration space \(Q\), one may define:

• A submanifold \({\mathcal M}^{nh}_q\) of \(Q\), which contains to the point \(q\) and whose dimension is equal to the rank of the constraint distribution, and
• A family of Riemannian metrics on \({\mathcal M}^{nh}_q\) such that the geodesics of every one of these metrics with starting point \(q\) are just the nonholonomic trajectories with the same starting point \(q\).

In particular, the previous facts imply that the kinetic nonholonomic trajectories with starting point \(q\), for sufficiently small times, minimize length in \({\mathcal M}^{nh}_q\)!

Hyperpolygons spaces are a family of hyperkähler manifolds that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we showed that these spaces are isomorphic to certain families of parabolic Higgs bundles, when a suitable condition between the parabolic weights and the spectra of the coadjoint orbits is satisfied. In analogy to this construction, we introduce two moduli spaces: the moduli spaces of quasi-parabolic \(SL(2,\mathbb{C})\)-Higgs bundles over \(\mathbb{C}\mathbb{P}^1\) on one hand and the null hyperpolygon spaces on the other, and establish an isomorphism between them. Finally we describe the fixed loci of natural involutions defined on these spaces and relate them to the moduli space of null hyperpolygons in the Minkowski 3-space. This is based on joint works with Leonor Godinho.

La teoría de campos de Chern-Simons es una teoría de campos topológica, y puede formularse sobre cualquier fibrado principal cuyo grupo de estructura admita una forma bilineal invariante. Consideremos un fibrado principal \(\pi:P\to M\) con grupo de estructura \(G\) y sea \(K\subset G\) un subgrupo; tomemos la familia de problemas variacionales de Chern-Simons sobre todos los subfibrados de \(P\) con fibra \(K\). En la presente charla explicaremos cómo esta familia de problemas variacionales puede codificarse mediante un único principio variacional de tipo Griffiths. Finalmente, utilizaremos el problema variacional construído para entender desde un punto de vista geométrico la correspondencia entre gravedad en dimensión \(2+1\) y la teoría de Chern-Simons.

Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective. This has opened up the possibility of introducing variational and symplectic integration methods using geometric integrators. In particular, in thistle, we introduce variational integrators which allows us to derive different methods for optimization. However, since the systems are explicitly time-dependent, the preservation of the symplecticity property occurs solely on the fibers. Finally, using discrete Lagrange-d'Alembert principle we produce optimization methods whose behavior is similar to the classical Nesterov method reducing the oscillations of typical momentum methods. Joint work with Cédric M. Campos and Alejandro Mahillo.

Los sistemas lagrangianos y hamiltonianos con fuerzas aparecen en distintos contextos tales como sistemas con fuerzas de control o sistemas con fuerzas de disipación y fricción. También es usual que los sistemas dinámicos que se obtienen al reducir una simetría de un sistema mecánico sean sistemas forzados. En los casos en los que las fuerzas no puedan ser absorbidas por el lagrangiano o el hamiltoniano como parte de un potencial, se realiza una descripción variacional alternativa del caso sin fuerzas. Como en el caso continuo, cuando se realiza un proceso de reducción de una simetría de un sistema mecánico discreto, es usual que el sistema dinámico reducido presentes términos que se pueden interpretar como fuerzas. Entre otras razones, esto hace que resulte interesante estudiar las características de los sistemas discretos forzados. Un caso particularmente interesante de reducción de simetrías de un sistema mecánico es el que se basa en el uso de sus cantidades conservadas. Cuando se considera un sistema mecánico discreto que admite una aplicación momento que se conserva sobre las trayectorias del sistema, se aplica el proceso conocido como la reducción de Routh discreta que da lugar a un sistema forzado. En esta charla se consideran sistemas mecánicos discretos forzados. A partir de su dinámica definida por un principio de Lagrange d'Alembert discreto que se define modificando convenientemente el principio variacional usual, se estudia la existencia de estructuras simplécticas y su posible conservación por la evolución del sistema. En particular, se considera el proceso de reducción de Routh discreta para una simetría no necesariamente abeliana de un sistema mecánico discreto como un caso particular de un proceso de reducción de simetrías más general. También se analizan la existencia y la conservación de estructuras simplécticas sobre el espacio reducido en el marco de los sistemas forzados teniendo en cuenta las características propias de este caso especial.

In this talk we present an extended version of the Hamilton-Jacobi equation (HJE), valid for general dynamical systems defined by vector fields (not only by the Hamiltonian ones), and a result which ensures that, if we have a complete solution of the HJE for a given dynamical system, inside a certain subclass of systems, then such a system can be integrated up to quadratures. Then we apply this result to show that the exponential curves \(\exp\left(\eta\,t\right)\) of any Lie group, which are the integral curves of the left invariant vector fields in the group, can be constructed up to quadratures (unless for certain elements \(\eta\) inside its corresponding Lie algebra). This gives rise to an alternative concrete expression of \(\exp\left(\eta\,t\right)\), different to those that appears in the literature for matrix Lie groups.

In this talk, I will start by explaining the nonhamiltonian nature of nonholonomic systems, and then we will study the "hamiltonization problem" from a geometric standpoint. By making use of symmetries and suitable first integrals of the system, we will explicitly define a new bracket on the reduced space codifying the nonholonomic dynamics that, in many examples, is a genuine Poisson bracket making the system hamiltonian (in general, the new bracket carries an almost symplectic foliation determined by the first integrals). This is a joint work with Luis Yapu-Quispe.