This is the subproject 3 of the coordinated project entitled Interacción Física-Tecnología-Matemáticas: métodos geométricos modernos, which is made of two more subprojects (at University of Zaragoza and University of La Laguna). The members of this subproject are mathematicians and physicists from the UPC. In this subproject we continue and enlarge our previous work on geometric aspects of dynamical systems in Mechanics, Control theory and its applications, and in Classical field theory and gravitation. We use methods of modern differential geometry and global analysis to advance in the solution of problems in these fields, to search for new models and tools to solve them, and to study the geometric structures that appear in such problems. The subjects and problems to be considered are: Classical field theories: To prove Darboux theorems for k-presimplectic and k-precosymplectic manifolds. To develop geometric constraint algorithms for singular k-presymplectic and k-precosymplectic classical field. To study the inverse problem using these structures and derivations along maps. To develop a framework for implicit Lagrangian systems (of PDEs) using Dirac structures and the k-symplectic and k-cosymplectic formulations. To study the correspondence between the unified formulations of the Einstein-Hilbert and the Einstein- Palatini models of gravity. To develop a multisymplectic formulation for the Lovelock and the f(R) models of gravitation and study their gauge and rigid symmetries, their conservation laws and reduction. To study the foundation of the asymptotic safety conjecture (ASC). To understand the final stages of black hole evaporation and resolve their instability in the framework of the ASC. To use the ASC to model quantum rotating black holes in a cosmological context. To apply the ASC to the description of our Universe, the resolution of the Big Bang singularity and the existence of dark energy. To study the foundations of the Hamilton-Jacobi method for describing the Hawking-Unruh radiation. To analyse the existence of a first law of the thermodynamics for marginally trapped surfaces. Geometric Mechanics: To state the Hamilton-Jacobi theory for time-dependent systems on fibre bundles and, as an application, study the Lamb equation. Control systems: To obtain a new geometrical description of hybrid systems in order to distinguish different families of control systems. To obtain a geometric formulation for including different kinds of control systems and state and solve problems related to their controllability, accessibility and optimal control. To find new and applicable conditions for differential flatness, and apply them to real systems. To apply previous algorithms for analyzing differential flatness to the quadrotor, the spherical robot and the free-floating robot, and obtain new algorithms and conditions using vector fields in Pfaffian systems. To study the consequences of adding or removing an actuator to a differentially flat system, generalizing some previous results. Geometric structures: To study Lie systems with compatible geometric structures (poli-symplectic, poliPoisson, multisymplectic). To generalize the coalgebra method for obtaining superposition rules, constants of motion and invariant tensors. To study the relation between multisymplectic and k-symplectic Lie systems. Reduction and reconstruction theory for Lie systems in multisymplectic manifolds.
De León, M.; Gaset, J.; Lainz, M.; Rivas, X.; Roman-Roy, N. Fortschritte der physik. Progress of physics Vol. 68, num. 8, p. 2000045-1-2000045-12 DOI: 10.1002/prop.202000045 Date of publication: 2020-06-23 Journal article