Some Aspects of
Symplectic structures arise in a natural way in theoretical mechanics, in
particular during the process of quantization, that is, in the passage from
classical to quantum mechanics. In order to motivate the study of symplectic
geometry, we will begin with a rough sketch of the relevant physics, although
we will not cover all the concepts of this field nor give all of the relevant
definitions. As references, one may consult the first chapter of
A more complete description of the principles of classical mechanics can be
[Wo], in the third chapter of
[A] and in the
third and fifth chapters of
[AM]. A further highly
recommendable classical source is the first chapter of
For the process of quantization, we refer the reader to §15.4 of KlRlLLOV
[Ki]. It is the goal of this text to later return and cover the topics of this
chapter in greater detail.
0.1. The Lagrange equations
The purpose of theoretical mechanics is the discovery of principles which
allow one to describe the time development of the state of a physical system.
In classical mechanics such a state is given as a point P on an n-dimensional
real manifold Q (see Section A.l). Q is called the configuration space, and
P is described by the local coordinates #i,... , qn, called position variables.
The time development of the system is then described by the curve 7 = P(t),
11—• P(t) with P(t0) = P°,