Exercícios de Física, Ana Noronha e Pedro Brogueira
McGraw-Hill, ISBN: 972-9241-72-4, 1994.

From the Preface

Este livro destina-se aos estudantes universitários que incluam a Física nos seus curricula e nasceu da nossa experiência de ensino nos primeiros anos das licenciaturas de IST. O objectivo é o de tornar claros os aspectos mais importantes das matérias versadas, através de um conjunto de exemplos e exercícios detalhadamente resolvidos, muitos dos quais foram propostos aos nossos alunos em exame. Cada capítulo contém além disso uma pequena introdução onde são apresentados os principais conceitos básicos, para referência rápida.
Acrescentámos ainda em Apêndice alguns conceitos básicos de Álgebra e Cálculo, introduzidos de maneira informal e através de pequenos exemplos de Física, a fim de construir uma referência rápida.

BIOMAT 2005, Proceedings of the International Symposium on Mathematical and Computational Biology, Rubem P. Mondaini and Rui Dilão
World Scientific Publishing, ISBN 981-256-797-6, 2006.


From the Preface

The BIOMAT 2005 International Symposium on Mathematical and Computational Biology, together with the Fifth Brazilian Symposium on Mathematical and Computational Biology,   was held in the city of Petrópolis, state of Rio de Janeiro, Brazil, from the 3rd to the 8th December 2005. The atmosphere of the symposium was informal and the approach interdisciplinary, with the contribution of the expertise of fifteen keynote speakers from different fields and  backgrounds.

In the proceedings of BIOMAT 2005, there are state of the art research papers in the mathematical modelling of cancer development, malaria and aneurysm development, among others. Models for the immune system and for epidemiological issues are also analyzed and reviewed. Protein structure prediction by optimization and combinatorial techniques (Steiner trees) are explored. Bioinformatics questions, regulation of gene expression, evolution, development, DNA and array modelling, small world networks are other examples of topics covered in the BIOMAT 2005 symposium.

The diversity of topics  and the combination of original with review approaches  make BIOMAT Symposia important events for graduate students and researchers.


BIOMAT 2006, International Symposium on Mathematical and Computational Biology, Rubem P. Mondaini and Rui Dilão
World Scientific Publishing, ISBN 981-270-768-9, 2007.


From the Preface

The BIOMAT 2006 Symposium was held in the city of Manaus in the Brazilian Equatorial Rain forest, from November the 25th to December the 1st.  We had fifteen Keynote Speakers from Europe and Americas and an impressive number of contributed works presented by scientists and research students from Brazil and abroad. The BIOMAT tutorials, which are already traditional in the BIOMAT symposia, and are lectured on the first two days of these conferences, are a source of motivation for future researchers in these interdisciplinary topics.


The topics of the BIOMAT 2006 Symposium were a combination of state of the art research and review approaches. They range  from cell dynamics and surface reaction models of protocells, to the study of collective steady states of cells, to the modelling of infectious diseases like HIV epidemiology, molecular

genetic mechanisms of hepatitis B virus, and the dynamics of tuberculosis. Models of physiological disorders like tumor growth and 3D reconstruction of objects were also analyzed.  Topics on the  modelling of DNA and proteins by using de novo structure prediction, substitution matrices and Steiner trees were discussed.  Other subjects covered in the BIOMAT 2006 Symposium were studies in population dynamics like insect sociality, multistability on predator-prey models, and techniques of impulsive differential equations in bio-economics.


Nonlinear Dynamics in Particle Accelerators, Rui Dilão and Rui Alves-Pires
World Scientific Series in Nonlinear Science, Series A, World Scientific Publishing, ISBN: 981-02-2517-2, 1996.


From the Preface

Particle accelerators are instruments of increasing importance in scientific research and industry, playing a prominent role in our understanding of the constituents of matter and their interactions. On the other hand, the great dimensions and resources involved in their construction and use have a significant public impact. The scientific research and the high degree of technological development  achieved in the construction of particle accelerators and associated equipment are stimulating the invention of new products and materials, the introduction of new methods of manufacturing, defining the frontiers of technology. 

This book is an introductory course to accelerator physics at the level of graduate students. It was written for a large audience which includes users of accelerator facilities, accelerator physicists and engineers, and undergraduates aiming to learn the basic principles of construction, operation and application of accelerators. It covers the  basic material for a one semester course in the physics and design of synchrotrons. In particular, chapter 1 is for the non-specialized reader.

In order to feel the main problems associated with the design of synchrotrons, we have included a software package to help the design of real synchrotrons, keeping control of the optical properties of beams. This software tool is based on the Mathematica programming language, and enables the reader to test its own skills in the design of synchrotrons. To guide the user in the optics of particle beams, we have included several examples of synchrotron design.

On the other hand, the new concepts of dynamical systems  developed in the last twenty years  give the theoretical setting to analyze the stability of particle beams in accelerators. In this book a common language to both accelerator physics and dynamical systems is integrated and developed, aiming to eliminate the difficulties faced by accelerator physicists, engineers and applied mathematicians when they try to join efforts in the attempt to improve the performance of synchrotron accelerators.

In chapter 1, after an historical introduction to the development of accelerators, the physical principles for their
design and construction are described. This chapter is self-contained and meets the needs of a simple course for the user of particle accelerator facilities. It was written at a very basic level.

In chapter 2, the equations of motion of a test particle in a synchrotron accelerator are derived. Transverse and
longitudinal motion  are treated separately. The physics of radio-frequency cavities is derived and the nonlinear Poincaré map for the longitudinal motion is analyzed. Several case studies of stable orbits in synchrotrons and the principle of alternating gradient focusing are analyzed.

In chapter 3, some basic concepts of the Theory of Dynamical Systems are introduced and  some of the results of the theory of nonautonomous differential equations and Poincaré maps are reviewed. Mathematical details are omitted and referred to specialized articles. The concepts of tune and dynamic aperture, familiar to the accelerator physicist, are explored in the context of dynamical systems where they have the meaning of rotation number and stability limit around a fixed point. The geometric aspects of chaotic behavior in conservative maps and homoclinic and heteroclinic explosions are explained. The concepts of linear, nonlinear and parametric resonance are discussed in detail, as they play an important role in the working point of a synchrotron. These results will be important in chapters 5 and 6 where the parameters of dynamically nonlinear synchrotrons are analyzed and predicted. 

In chapter 4, the Courant and Snyder linear theory of the transverse motion of particles in synchrotron accelerators is presented and its stability analyzed. These methods are the main tools for accelerator design. The  thin-lens approximation is introduced and the general Poincaré map for the transverse motion  is derived.

Basic techniques of the theory of dynamical systems are presented in chapter 4 and are used to analyze beam aberrations and  to calculate the chromatic correction. We obtain the general formula for the chromaticity parameter, which is critical in the choice and location of sextupoles in large synchrotrons.The problems of momentum correction, field errors and imperfections are analyzed in detail.

With the  tools developed in chapter 4, a simple computer program was developed by the authors to design synchrotrons. The code uses the energy, characteristics of magnets and straight sections as input parameters. This program determines the stability of the design orbit, calculates chromaticities, Poincaré maps, tunes, slip factor and the popular Courant and Snyder beta, alpha and gamma functions, and dispersion.

In chapter 5, the Poincaré  map for a synchrotron with nonlinear magnetic fields is derived. Based on the theory of dynamical systems exposed in chapter 3, nonlinear  optimization strategies for the strength of sextupoles and tune shift are derived. The problem of nonlinear tune resonances and dynamic aperture is analyzed and discussed in detail. This material corresponds to previous published work of the authors. In order to compare theoretical results with real design situations we have compared our predictions with the design simulations of the Large Hadron Collider under design and study at CERN. The predictions obtained agree within a relative error of the order of 1%.

Chapter 6 concerns the study of the beam-beam interaction.Stability conditions, tune and tune shift values for the nonlinear beam-beam map are derived.The dimensions of the chaotic annular region around the central design orbit are estimated. 

Table of Contents


1 Introduction to accelerator physics    1 

1.1 Introduction  1

1.2 Accelerators in medicine and industry  4

1.3 Nonlinear phenomena in particle accelerators  6

1.4 Acceleration and guiding of charged particles. Physical principles  7

1.5 Accelerator types  10

1.5.1  Linear accelerators  10

1.5.2 The cyclotron  11

1.5.3 The isochronous cyclotron  11

1.5.4 The synchrocyclotron  11

1.5.5 The betatron  12

1.5.6 The synchrotron  12

1.6 How to build a synchrotron accelerator  13

1.6.1 Confinement  13

1.6.2 Acceleration  17

1.6.3 Focusing  20

1.7 Synchrotron radiation  24

1.8 A short guide to accelerator bibliography  27


2 Equations of motion    29

2.1 Equations of motion in the transverse plane    30

2.1.1 Transverse motion in straight sections  32

2.1.2 Transverse motion in straight sections with special purpose magnets  34

2.1.3 Transverse motion inside dipoles  38

2.2 The design orbit and its stability   41

2.3 Focusing properties of pairs of quadrupoles  48

2.4 Longitudinal motion  52


3 Introduction to the qualitative theory of nonlinear differential equations   63

3.1 Phase space, fixed points and stability  65

3.2 Floquet theory  74

3.3 Poincaré maps  78

3.4 Maps of the plane. Homoclinic and heteroclinic orbits  82

3.5 Normal forms for two-dimensional conservative maps  90

3.6 Tune, tune shift and resonance  98


4  Dynamics and stability of guiding and focusing. Linear optics of synchrotrons    103

4.1 Transfer matrices, Poincaré maps and integrability  of the equations of motion  104

4.2 The Courant and Snyder $\beta$-function  107

4.3 Invariants, emittance, beam envelope and admittance  115

4.4 Simple examples of synchrotron design  121

4.5 Off-momentum motion. The dispersion function  126

4.6 Focusing-defocusing (FD) and focusing-bending-defocusing (FBD) basic cells  130

4.6.1  The FD cell  132

4.6.2 The FBD cell  136

4.7 Chromaticity and chromatic correction  138

4.8  Linear field errors  145

4.9 The design of a synchrotron  147


5 Nonlinear motion in the transverse plane: sextupoles   153

5.1 Nonlinear equations of motion in the thin lens approximation  155

5.2 Tracking simulations  160

5.3 Chromatic correction  167

5.4 Beam lines with one sextupolar error  170

5.5 Coupled motion in the two transverse directions  172


6 The beam-beam interaction   175

6.1 Equations of motion for the beam-beam effect  177

6.2 The general Poincaré map for localized interactions  178

6.3 The beam-beam force  181

6.4 Dynamics of the linearized beam-beam Poincar\'e map  182

6.5 The nonlinear beam-beam  map  186

6.6 Beam-beam tune shift  191



A1 Motion of relativistic charged particles in uniform electromagnetic fields  195

A2 An accelerator design program  199

A3 The $\bar p$ accumulator design  209


References 215


Subject index  221


Proceedings of the 2005 International Symposium on Mathematical and Computational Biology, BIOMAT 2005, Rubem P. Mondaini and Rui Dilão

E-papers servi\c cos editoriais, Rio de Janeiro, 2006.  ISBN 85-7650-064-7.

From the Preface

We had participants and contributors from many parts of the world as well as from almost all the brazilian states. The atmosphere of interdisciplinary work as driven by the expertise of our keynote speakers corresponds to the fundamental aims of the conference.


BIOMAT 2007, International Symposium on Mathematical and Computational Biology, Rubem P. Mondaini and Rui Dilão
World Scientific Publishing, ISBN: 981-281-232-2, 2007.


From the Preface

The present volume contains the contributions of the keynote speakers of the BIOMAT 2007 Symposium as well as selected contributed papers in the areas of mathematical biology, biological physics, biophysics and bioinformatics. It contains new results on some aspects of Lotka–Volterra equations, the proposal of using differential geometry to model neurosurgical tools, recent data on epidemiological modeling, pattern recognition and comprehensive reviews on the structure of proteins, the folding problem and the influence of Allee effects on population dynamics.

Onde Estás?, Materiais para Observar e Experimentar, Rui Dilão

Ministério da Ciência e da Tecnologia, Agência Ciência Viva, 1999

ISBN: 972--97805--1--X. Versão inglesa: ISBN: 972--97805--7--9.

From the Preface

This  booklet describes several experiments and observation activities aimed  at children aged eight and over. The activities may be used as a stimulus to observation by parents or by teachers in Basic Education. The booklet comes with a compass and five cut-outs to make a globe, a sun dial, a protractor, a quadrant and a nocturnal. Building and using these instruments act as a stimulus to  observe the world around us. The booklet is completed by the Latitude and Longitude text.

Latitudes e Longitudes, Rui Dilão

Ministério da Ciência e da Tecnologia, Agência Ciência Viva, 1999

ISBN: 972--97805--2--8. Versão inglesa: ISBN: 972--97805--8--7.

From the Preface

This booklet continues the experiments and activities introduced in Where are you? Material for observing and experimenting, and is aimed at children aged ten and over.  The experiments described here together with the instruments described in the first booklet can be used to carry out experimental activities and observation at a basic education level. You should read the first booklet and carry out the activities and experiments described there before you read this one.

Termodinâmica e Física da Estrutura da Matéria, Rui Dilão
Escolar Editora, ISBN: 978-972-592-317-7, 2011. 2ª edição 2014.

From the Preface

Neste texto, aborda-se a termodinâmica dos processos reversíveis, os princípios da termodinâmica do equilíbrio, os fundamentos  da teoria cinética dos gases, o conceito de entropia  e alguns aspectos da física moderna. Estes temas correspondem ao curriculum mínimo de termodinâmica e de física da estrutura da matéria de qualquer curso de engenharia e de física e pode ser seguido por alunos do primeiro ano da universidade.

Para a maioria dos cursos de engenharia e de física, a matéria abordada  é suficiente para ter uma visão qualitativa e quantitativa dos  fenómenos que envolvem trocas de calor e máquinas térmicas.  As máquinas térmicas que estão presentes no nosso  dia-a-dia são os motores dos automóveis e dos aviões, os frigoríficos, os sistemas de ar condicionado, as centrais de produção de energia, os seres vivos e o sistema climático global.

Introduzem-se alguns dos aspectos mais relevantes da Física do século XX, como sejam a descoberta da natureza corpuscular da luz e a descoberta da natureza ondulatória da matéria. Como muitos dos resultados da física moderna fogem à  intuição do dia-a-dia, optou-se por expor estas matérias tendo como base resultados experimentais.


Técnicas Matemáticas da Física, Rui Dilão
IST Press, ISBN: 978-989-8481-73-3, 2019.

From the Cover

Este texto é uma introdução aos fundamentos e linguagem da matemática da física moderna e da engenharia, desenvolvidos desde o início do século xx. Os conceitos mais importantes abordados neste curso são a teoria do integral de Lebesgue, os espaços de Hilbert, a teoria dos operadores, a análise de Fourier, a teoria das distribuições ou funções generalizadas e as equações às derivadas parciais. Estas matérias estão presentes em todas as abordagens da física ao mundo macroscópico e microscópico. Da mecânica quântica e teoria quântica dos campos aos aspetos mais modernos da teoria das cordas, o formalismo e a linguagem que aqui se introduzem são essenciais para a compreensão da física atual. Do ponto de vista das aplicações, a eletrónica dos semicondutores, a estrutura interna dos microprocessadores, o laser, os ecrãs planos, entre outros, são consequências práticas da física moderna.

Este livro baseia-se no curso de Técnicas Matemáticas da Física lecionado no terceiro ano do Mestrado Integrado em Engenharia Física Tecnológica do Instituto Superior Técnico.  Um dos seus objetivos foi o de tornar acessível, nos primeiros anos da universidade, resultados matemáticos muito poderosos, permitindo uma abordagem pragmática à física moderna. Abandonou-se a formulação tradicional da matemática na sua estrutura tradicional de lema/teorema, optando-se por uma formulação mais discursiva, sem perder a exatidão dos resultados.

Este livro destina-se aos alunos universitários de graduação e de pós-graduação que queiram desenvolver bases sólidas nas técnicas modernas de modelação matemática e computacional, no estudo da mecânica quântica e da teoria dos campos. As matérias abordadas integram os currículos básicos dos cursos de Física, Matemática e Engenharia Eletrotécnica e Mecânica.