# About GMC Network

# Introduction

The research on mechanical and dynamical systems has had a deep impact in other research areas and in the development of several technologies. A big part of its advances has been based on numerical and analytical techniques. In the sixties, the most sophisticated and powerful techniques coming from Geometry and Topology were used in the study of dynamical systems. Those techniques led, for instance, to the beginning of the modern Hamiltonian Mechanics. Geometric techniques have been also applied to a wide range of control problems such as locomotion systems, robotics, etc. Most of these ideas have been developed in the last 30 years by mathematicians of a high scientific level such as J. Marsden, A. Weinstein, R. Abraham, V. Arnold or R. Brockett among others.

The emphasis on geometry means an attempt of understanding the structure of the equations of motion of the system in order to analyze them and study their design. The symplectic structures play a fundamental role in the differential-geometric description of the Lagrangian and Hamiltonian Mechanics on the tangent and cotangent bundle of the configuration space. In particular, it is important in the study of “regular systems” (a notion linked to the non-degeneracy of the symplectic form). The symplectic geometry allows us to obtain, starting from the Hamiltonian energy and in a simple way the dynamics (the Hamiltonian vector field), whose integral curves satisfy the Hamilton’s equations. When the system is singular (the Lagrangian function is not regular), that is, when there exist internal constraints, we must replace the symplectic structure for a more general geometric object: Poisson structures. These objects also play an essential role in the geometric description of the reduction of mechanical systems which admit a Lie group of symmetries. It is particularly interesting the case of linear Poisson structures on vector bundles. There are other singular systems in which the constraints are external: the non-holonomic mechanical systems. The dynamics for this kind of systems is controlled by a “quasi-Poisson” structure, a geometric object with the same properties as a Poisson bivector, but the integrability condition is not satisfied. For all these topics, the members of the network have made relevant contributions along the last 25 years.

A category which is closely related with Poisson Geometry is that of Lie algebroids. A Lie algebroid is a natural generalization of the tangent bundle and of the Lie algebras. On the other hand, there exists a one-to-one correspondence between Lie algebroid structures on a vector bundle and linear Poisson bivectors on the dual bundle. As a consequence, it is not surprising the usefulness of Lie algebroids in the geometric description of Mechanics. This fact was first noted by A. Weinstein and, after him, it has been recently studied by several authors, some of them belonging to this network.

During the last decade, the construction of geometric integrators for Lagrangian systems using discrete variational principles, has been a topic of growing interest. Of a particular interest we find systems with discrete Lagrangian over the cartesian product of the configuration manifold with itself. This cartesian product is the discrete version of the phase space of velocities. When the discrete Lagrangian is an approximation of a continuous Lagrangian function, it is obtained a geometric integrator which inherits some properties of the continuous Lagrangian. Some extensions of these results with the purpose of construct geometric integrators for more general mechanical systems, such as non-holonomic or time-dependent systems, have been obtained by several members of this network.

On the other hand, Moser and Veselov have recently considered discrete Lagrangian systems on Lie groups, what later motivated A. Weinstein to start the study of discrete Mechanics on Lie groupoids. Lie groupoids are a natural generalization of Lie groups and cartesian products of a manifold by itself and, moreover, Lie algebroids can be considered as the infinitesimal version of Lie groupoids. The work started by A. Weinstein, as well as some other problems arisen by him, are nowadays a research topic to some members of this network.

Classical Mechanics can be considered as a Classical Field Theory of first order in which space-time has dimension one. As it is well-known, the space-time of a Classical Field Theory of first order is a smooth manifold and the dynamics is given as the solutions of a partial differential equation. One of the geometric formalisms, broadly accepted, which is used in the description of Classical Field Theories since the seventies, is the multisymplectic approach. Pioneer works in this sense were made by some of the members of this network. The notion of a multisymplectic structure became the key notion to definitely clarify the Hamiltonian Field Theory, which had been thoroughly treated in previously decades. This doctrine is an important interdisciplinary area whose many aspects of research are far away from being finished. In fact, nowadays it is one of the research lines of an extensive group of members of the network. To this fact we have to add a renovated interest in Multisymplectic Geometry partly due to the discovery of numerical integrators which preserve the multisymplectic form. In fact, some members of this network are starting some investigations in this line of research.

All the lines of research previously mentioned are part of a wider research topic generically called Geometric Mechanics, which is a meeting point for several disciplines such as Geometry, Analysis, Algebra, or Partial Diferential Equations. Geometric Mechanics is a growing research area with fruitful connections with other disciplines such as Nonlinear Control Theory or Numerical Analysis.

Starting in the seventies, Diferential Geometry has played an important role in the development of Control Theory. Since then, several questions such as controlability, observability, dynamical reaction, linearization and stabilization of the reaction or optimal control, are linked to geometric concepts such as Lie group theory, differential forms, distributions, homogeneous spaces, symplectic structures, Sub-Riemannian geometry, etc. In fact, some of the developments of Diferential Geometry are a consequence of the geometrization of control problems, creating a mutual benefit between both disciplines. Recently, several members of this network project has been interested by the application of geometric techniques to optimal control theory, in particular to obtain numerical integrators for optimal control problems.

The organization of this network is based in on-going collaborations that have already been materialized in publications and joint activities between the different groups being part of this network. This fact guarantees the feasibility of the project, which incorporates at the same time some young researchers, who have received an excellent preparation in well-known research centers and who proportionate the network a guaranty of continuity for future activities.

The members of this Thematic Network project center their work around these axis: Geometry, Mechanics and Control, all clearly interconnected.

# Topics

### Geometry

- Symplectic, Poisson and Jacobi manifolds
- Lie group geometry in classical and quantum mechanics
- Physics in spaces with constant curvature
- Cayley-Klein geometries: models and applications in Physics
- Geometry of gauge theories

### Mechanics

- Non-holonomic mechanics
- Vakonomic dynamics
- Calculus of variations
- Systems and conservation laws. Noether theorems
- Classical field theories and multisymplectic geometry
- Numeric-geometrical integration of mechanical systems
- Lie algebroids and geometric mechanics
- Lie groupoids and discrete mechanics
- Integrable and super-integrable systems
- Geometric quantization
- Symmetries. Lie group actions. Reduction of order
- Relative equilibria and relative periodic orbits in symmetric Hamiltonian systems. Stability and bifurcations

### Control Theory

- Optimal control theory
- Control of mechanical systems
- Distributed control
- Geometric control

# Objectives

- Facilitate the exchange and transference of knowledge between the Spanish groups which work in different aspects related with Geometric Mechanics, Field Theory and Control Theory.
- Define research lines as a result of the group activities among the different members of the network and organize the research in every group more precisely and in collaboration with the other groups.
- Promote the collaboration among different reseach groups both in European and international networks. There already exist collaborations with several groups of these networks, what will definitely help to improve this cooperation. It is specially interesting the identification of a group in a topic which could play a Spanish reference role in the European Framework of Research.
- Give to the new generations of researchers a common framework for their apprenticeship in aspects related with the research in Geometric Mechanics, Field theory and Control theory.