1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE l. REDUCTION THEORY

Let the Gelfand-Fuchs cocycle be defined by^2

L:(u,v) = / fo

1

u'(x)v"(x)dx,

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where / is a constant. Let the Virasoro Lie algebra be defined by g = g x JR

with the Lie bracket

[(u, a), (v,b)] = ([u,v],1L:(u,v)).

This is verified to be a Lie algebra; the corresponding group is called the Bott-

Virasoro group. Let

l(u, a)= -a^1 2 +^11 u^2 (x)dx.

2 0

Then one checks that the Euler-Poincare equations are

da

dt

du

dt

0

• 1au^111 - 3u' u

so that for appropriate a and / and rescaling, we get the KdV equation. Thus, the

KdV equations may be regarded as geodesics on the Bott-Virasoro group.

Likewise, the Camassa-Holm equation can be recast as geodesics using the H^1

rather than the L^2 metric (see Misiolek [1997] and Holm, Kouranbaeva, Marsden,

Ratiu and Shkoller [1998]).

Lie-Poisson Systems. The Euler-Poincare equations occur on g , while the Lie-

Poisson equations occur on g*. To understand how one arrives at g* we recall a

few facts about reduction.

If P is a Poisson manifold and G acts on it freely and properly, then P / G is also

Poisson in a natural way: identify functions on P /G with G-invariant functions on
P and use this to induce a bracket on functions on P/G. In the case P = TG and
G acts on the left by cotangent lift, then TG/G ~ g* inherits a Poisson structure
given explicitly by the following theorem.

Theorem 1.3. (Lie-Poisson reduction.) The Poisson structure inherited on g*

is given by

{f,g}-(μ) = -\μ, [~~' ;~])'

the Lie-Poisson bracket. For the right action, use+.

If H is G-invariant on T*G and XH is its Hamiltonian vector field determined

by p = {F, H}, then XH projects to the Hamiltonian vector field xh determined

by j = {!, h} where h = HIT;G = Hlg*. We call j = {!, h} the Lie-Poisson

equations.

If l is regular; i.e.' ~ f-+ μ = az I 8~ is invertible, then the Legendre transforma-

tion taking ~ to μ and l to

h(μ) = (~, μ) - t(O

(^2) An interesting interpretation of the Gelfand-Fuchs cocycle as the curvature of a mechanical
connection is given in Marsden, Misiolek, Perlmutter and Ratiu [1998a,b].