This book was based on notes which were prepared as a guide for lectures of one semester course on Geometric Mechanics. They were written inside the level of a master course. I started some years ago teaching them at the "Instituto de Matematica e Estatistica" of the "Universidade de Sao Paulo", and, more recently, at the "Instituto Superior Tecnico" of the "Universidade Tecnica de Lisboa".
The spectrum of participants of such a course ranges usually from young Master students to Phd students. So, it is always very difficult to decide how to organize all material to be taught. I decided that the expositions should be self contained, so some subjects that one expects to be interesting for someone, result, often, tedious for others and frequently unreachable for a few ones.
In any case, for young researchers interested in differential geometry and or dynamical systems, it is basic and fundamental to see the foundations and the development of classical subjects like Newtonian and Relativistic Mechanics.
I wish to thank a number of colleagues from several different Institutions as well as Master and PhD students from Sao Paulo and Lisbon who motivated and helped me with comments and suggestions when I was writing this text. Among them I mention Jack Hale, Ivan Kupka, Giorgio Fusco, Paulo Cordaro, Carlos Rocha, Luis Magalhaes, Luis Barreira, Esmeralda Dias, Za-queu Coelho, Helena Castro, Marcelo Kobayashi, Sonia Garcia, Diogo Gomes and Jose Natario. 1 am also very grateful to Ms. Aclu Dosanjh of Springer-Verlag for her help and encouragement; it has been a pleasure working with her and her Springer-Verlag colleagues. Thanks are also due to Ana Bor-dalo for her fine typing of this work and to FCT (Portugal) for the support through the program POCTI.