Symplectic Reduction

2)-dimensional
. . ,such
24 Reduction
24.1 Noether Principle
Let (M, w, G, p) be a hamiltonian G-space.
Theorem 24.1 (Noether) If f : M -+ R is a G-invariant function, then p
is constant on the trajectories of the hamiltontan vector field of f.
Proof. Let vf be the hamiltonian vector field of f. Let X E g and px
(y, X) : M -+ R. We have
'C'f /-tx =
zvf dlix = zvf zx*w
=
-%X*zvfw=-zx*df
=
-,CX*f 0
because f is G-invariant.
Definition 24.2 A G-invariant function f M -+ R is called an integral of
motion of (M, w, G, p). If p is constant on the trajectories of a hamiltonian
vector field vf, then the corresponding one-parameter group of diffeomorphisms
exp tvf I t E R} is called a symmetry of (M, w, G, p).
The Noether principle asserts that there is
between symmetries and integrals of motion.
a one-to-one correspondence
24.2 Elementary Theory of Reduction
Finding a symmetry for a 2n-dimensional mechanical problem may reduce it
to a (2n
-
problem as follows: an integral of motion f for a
2n-dimensional hamiltonian system (M, w, H) may enable us understand the
trajectories of this system in terms of the trajectories of a (2n
-
2)-dimensional
hamiltonian system Wred) Wred) Hred). To make this precise, we will describe
this process locally. Suppose that U is an open set in M with Darboux coordinates
x,.... I X"' 1' that
. f for this chart, and write H in these
coordinates: H = H (x 1, x, Then
.
.,
is an integral of motion
the trajectories Of VH lie on the
hyperplane
J , HI 0 =
H
= constant
OH