XIV
Preface
which corresponds precisely to that of the symplectic manifolds. On the
other hand, both may be viewed as pre-symplectic manifolds. Here the
connection will be given through the example of a contact manifold as the
surface of constant energy of a Hamiltonian system.
The fourth and fifth chapters will be a mix of further mathematical
constructions and their physical interpretations. This will begin with the
description of the moment map attached to the situation of a Lie group G
acting symplectically on a symplectic manifold such that every Hamiltonian
vector field is global Hamiltonian. This is a certain function
The most important examples of the moment maps are the Ad*-equivariant
ones, that is, those that satisfy a compatibility condition with respect to the
coadjoint representation Ad*. The first result of Chapter 4 is that for a sym-
plectic form uu = —dfi and a G invariant 1-form i? such an Ad*-equivariant
map can be constructed. This will then be applied to the cotangent bundle
T*Q, as well as to the tangent bundle TQ, where it will turn out that for
a regular Lagrangian function L G F(Q) the associated moment map is an
integral for the Lagrangian equation associated to L. As examples, we will
discuss the linear and angular momenta in the context of the formalism of
the moment map, and so make clear the reason for this choice of terminology.
Next, we describe symplectic reduction. Here, we are given a symplectic
G-operation on M and an Ad*-equivariant moment map t ; under some
relatively easy-to-check conditions, for fi G 0*, the quotient
is again a symplectic manifold. This central result of Chapter 4 has many
applications, including the construction of further examples of symplectic
manifolds (in particular, we obtain other proofs that the projective space
Pn(C)
as well as the coadjoint orbits are symplectic). Another application is
the result of classical mechanics on the reduction of the number of variables
by the application of symmetry, leading to the appearance of some integrals
of the motion.
In the fifth and last chapter, we consider quantization', that is, the tran-
sition from classical mechanics to quantum mechanics, which leads to many
interesting mathematical questions. The first case to be considered is the
simplest: M
R2n
=
T*Rn.
In this case the important tools are the groups
SX2O&), 5p2n(lK0, the Heisenberg group Heis2n(IR) the Jacobi group G ^ R )
(as a semidirect product of the Heisenberg and symplectic groups) and their
associated Lie algebras. It will follow that quantization assigns to the poly-
nomials of degree less than or equal to 2 in the variables p and q of
R2n
an operator on
L2(M)
with the help of the Schrodinger representation of
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