I like to think of symplectic geometry as playing the role in mathematics of a language
which can facilitate communication between geometry and analysis. On the one hiind, since
the cotangent bundle of any manifold is a symplectic manifold, many phenomena and con-
structions of differential topology and geometry have symplectic "interpretations", some of
which lead to the consideration of symplectic manifolds other than cotangent bundles. On
the other hand, the category of symplectic manifolds has formal similarities to the categories
of linear spaces used in analysis. The problem of constructing functorial relations which re-
spect these similarities is one aspect of the so-called
problem; using the solu-
tions of this problem which are presently available, one can construct analytic objects (e.g.,
solutions of partial differential equations, representations of groups) from symplectic ones.
The most important objects in symplectic geometry, after the symplectic manifolds
themselves, are the so-called
submanifolds. They arise in a multitude of ways in
the symplectic interpretation of geometric phenomena; on the other hand, they correspond
under quantization relations to elements (or classes of elements) of the linear spaces of
The oldest symplectic manifold of my acquaintance occurs in Lagrange's later work
(1808) on celestial mechanics [LA 1] . Lagrange wrote the equations of notion for the orbit-
al elements
, ... ,
of a planet, under the effect of perturbations, in the form
where a;;(x) is a 6 by 6 skew-symmetric matrix, and he showed that a
suitable choice of the x·coordinate system puts these equations in the form now known as
Hamilton's equations. Indeed, the development of symplectic geometry is closely connected
with developments in classical mechanics; nevertheless, since there are several extensive
studies available on symplectic mechanics ([AB], [AN 4], [AN V], (SI], [SR]), I shall
largely neglect classical mechanics and dynamical systems in the present lectures. (When
physics appears, it will usually be quantum mechanics.) We also recommend to the reader
the book [GU), which discusses in great detail many topics which are merely men-
tioned here.
The first six sections of these notes contain a description of some of the basic construc-
tions and results on symplectic manifolds and lagrangian submanifolds. §7, on intersections
of lagrangian submanifolds, is still mostly internal to symplectic geometry, but it contains
some applications to mechanics and dynamical systems. §§8, 9, and 10 are devoted to vari-
ous aspects of the quantization problem. In §10 there is a feedback of ideas from quantiza-
tion theory into symplectic geometry itself.
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