Integrability, non-integrability, chaos and control in classical and quantum mechanics

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             University of Zielona Góra              Institute of Physics                 Institute of Astronomy       Center for Theoretical Physics, PAS

AstroNet

AstroNet is a Marie-Curie research training network on astrodynamics that that brings together mathematicians, engineers and astronomers from universities, governmental agencies and industry. The network continues the training and research undertaken by the former training network AstroNet (2007–10).

The research topics of the network include innovative new methods for designing spacecraft trajectories and controlling their dynamics. Particular emphasis is placed on optimizing trajectories and control to minimize fuel use and extend mission ranges. This is achieved by maximizing the use of natural dynamics and employing sophisticated ideas and techniques from dynamical systems theory. The results are being extended to studies of the dynamics and control of novel spacecraft architectures, such as solar sails, space tethers and formations of spacecraft.

The scientific objectives of the research programme are grouped in three sections.

In this project eleven research centres participate.

  1. Trajectory design and control. The aim of this research is to study models that can be used to design energy-efficient spacecraft trajectories. The underlying philosophy is to exploit the natural dynamics of a spacecraft moving in the solar system. The subfields of the area are: 1.1 Invariant manifold dynamics, 1.2 Low thrust non-Keplerian orbits, 1.3 Coupled attitude-orbit dynamics and 1.4 Asteroid and binary asteroids missions.

  2. Attitude control and structural flexibility of spacecraft. The current spacecraft attitude control algorithms make little use of recent advances, such as geometric mechanics, control theory and complexity science. The overall aim of this section is to develop, analyze and test mathematical models and algorithms for a variety of different attitude control mechanisms using the aforementioned new techniques. The subfields of this area are: 2.1 Optimal and feasible attitude motions for micro-spacecraft, 2.2 Modelling and attitude control of flexible spacecraft, and 2.3 Dissipative effects on attitude dynamics.

  3. Formation flying. There is a limited volume of research into manoeuvring without collisions in enviroments other than free space. Furthermore, none of the existing research has looked at fully decoupling the relative motion and exploiting the natural dynamics of the suitable models to ensure safe formation reconfiguration. The basic aim of this section is to study formation flying manoeuvers under the aforementioned constraints and models. The subfields of this area are: 3.1 Transfers, deployment and proximity manoeuvring for multiple spacecraft, 3.2 Formation flying using low-thrust propulsion, and 3.3 Space inspection and autonomy.

  1. Trajectory Design and Control. The aim of this section is to study models that can be used to design energy-efficient spacecraft trajectories. The underlying philosophy is always to exploit the natural dynamics of a spacecraft moving in the solar system. The subsections of the item are: 1.1 Invariant Manifold Dynamics, 1.2 Low Thrust Non-Keplerian Orbits, 1.3 Coupled Attitude-Orbit Dynamics and 1.4 Asteroid and Binary Asteroids Missions.

  2. Attitude Control and Structural Flexibility of Spacecraft. The current spacecraft attitude control algorithms make little use of recent advances such as geometric mechanics, control theory and complexity science. The overall aim of this section is to develop, analyse and test mathematical models and algorithms for a variety of different attitude control mechanisms, using the above mentioned new techniques. The subsections of this item are: 2.1 Optimal and Feasible Attitude Motions for Micro-Spacecraft, 2.2 Modelling and Attitude Control of Flexible Spacecraft and 2.3 Dissipative Effects on Attitude Dynamics.

  3. Formation flying. There is a limited volume of research into manoeuvring without collisions in enviroments other than free space. Furthermore, none of the existing research has looked at fully decoupling the relative motion and exploiting the natural dynamics of the suitable models to ensure safe formation reconfiguration. The basic aim of this section is to study formation flying manoeuvers under the aforementioned constraints and models. The subfields of this area are: 3.1 Transfers, deployment and proximity manoeuvring for multiple spacecraft, 3.2 Formation flying using low-thrust propulsion, and 3.3 Space inspection and autonomy.

In this project eleven research centres participate.

  • Institut d'Estudis Espacials de Catalunya, Barcelona (Spain)&emdash;coordinator;

  • Clyde Space Limited, Glasgow (United Kingdom);

  • Deimos Space SL, Madrid (Spain);

  • GMV Aerospace and Defence SA, Madrid (Spain);

  • Middle East Technical University, Ankara, Turkey;

  • Politecnico di Milano, Milano, Italy;

  • University of Strathclyde, Glasgow, United Kingdom.;

  • University of Surrey, Guildford, United Kingdom;

  • Turun Yliopisto, Turku, Finland

  • Universita degli Studi di Roma Tor Vergata, Roma, Italy.;

  • Uniwersytet Zielonogorski, Zielona Góra, Poland.

The main interests of the Zielona Góra working group are in the area of Trajectory design and control, and in particular Asteroid and binary asteroid missions.

Read more about this project on the official web site

 

 

Maestro

The main aim of the project is a systematic analysis of solvability, integrability and control in quantum mechanics. The fundamental problem of solvability/integrability was raised from the beginning of quantum mechanics and now many solvable systems have been found by various techniques, usually based on specific properties of a given system, e.g., its symmetries suggesting a certain clever Ansatz for general solutions.

For classical systems solvability (in quadratures) is related directly to such notions as

1. integrability, understood as the existence of an appropriate number of conserved quantities (first integrals, symmetries, etc.),

 2. control problems for systems, or, more precisely, for Hamiltonians governing them,

 3. the absence of chaos.

 The quantum counterparts of these notions cause some fundamental problems already at the level of their definitions, but solvability itself seems to be the most amenable to definition for the quantum case and further to become a solid basis for the manifold concepts listed above. For this reason, it is very important to make solvability analysis as systematic as possible and perform it for wide classes of quantum systems.

We aim at obtaining general results and gaining a better understanding of the general mechanisms related to quantum solvability. To achieve this goal, we propose a novel approach to the solvability problem. Equations of quantum mechanics are linear differential equations with variable coefficients, possibly in an infinte-dimensional Hilbert space. To analyze solvability of such equations the powerful tools of differential algebra seem to be the most natural, but they have not been used widely enough in previous investigations of this topic.

 The project consists of researchers and students from various institutions: The Center for Theoretical Physics of the Polish Acedemy of Science (host institution, head Prof. dr hab. M. Kuś): Faculty of Physics and Astronomy of University of Zielona Góra; Institute of Physics of N. Copernicus University and Institut National des Sciences Appliquées de Rouen. At every of these institutions some Postdoctoral Research Posts and/or PhD positions are available.'

Read more about this Project on the official website

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