This book describes, by using elementary techniques, how some
geometrical structures widely used today in many areas of physics, like
symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics.
It is assumed that what can be accessed in actual experiences when
studying a given system is just its dynamical behavior that is described
by using a family of variables ("observables" of the system). The book
departs from the principle that ''dynamics is first'' and then tries to
answer in what sense the sole dynamics determines the geometrical
structures that have proved so useful to describe the dynamics in so
many important instances. In this vein it is shown that most of the
geometrical structures that are used in the standard presentations of
classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian,
Lagrangian) are determined, though in general not uniquely, by the
dynamics alone. The same program is accomplished for the geometrical
structures relevant to describe quantum dynamics. Finally, it is shown
that further properties that allow the explicit description of the
dynamics of certain dynamical systems, like integrability and super
integrability, are deeply related to the previous development and will
be covered in the last part of the book. The mathematical framework used
to present the previous program is kept to an elementary level
throughout the text, indicating where more advanced notions will be
needed to proceed further. A family of relevant examples is discussed at
length and the necessary ideas from geometry are elaborated along the
text. However no effort is made to present an ''all-inclusive''
introduction to differential geometry as many other books already exist
on the market doing exactly that. However, the development of the
previous program, considered as the posing and solution of a generalized
inverse problem for geometry, leads to new ways of thinking and relating
some of the most conspicuous geometrical structures appearing in
Mathematical and Theoretical Physics.
