1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 371

Thus, dG = -D is the symplectic form. By Stokes theorem,

1

e + J e + J e = f de ,

co c 1 c2 } S3

where Si are surfaces that project to the cap I: shown in the figure. (A more careful

argument-as in holonomy theorems-shows that the existence of these surfaces is

not necessary.)

Noting that

( D = (Dr'

l s; JE

where Dr is the reduced symplectic form (the symplectic form-an area

element-on the three wave surface), we find

1

8 + 1 8 + 1 8 = - ( Dr.

Co C1 c2 JE

On the group curves Ci we have

i = 1 ,2,3 where K i is the (constant) conserved quantity and </>i is the phase shift
associated with the ith S^1. This is b ecause K i is homogeneous of deg1ee 2.
On the dynamic trajectory co we have

1

8 = 1 (8 , X H) dt = ~HT

co co 2

where H is the constant energy of the trajectory and T is the period of the reduced
trajectory. This is because H is homogeneous of degree 3 in the qi.
Letting A(L:) denote the symplectic area of I: and putting the above equations
together, we obtain a phase identity:

3

2HT - </>1K1 - </>2K2 = -A(L:).

An important special case occurs on the fixed point space of the interchange

symmetry q 1 t-t q 3 in the case that /'I = ')' 3. This corresponds to the case of second

harmonic generation in nonlinear optics. In this case, from the formulas for the
two points P 1 and P 2 , we see that we can assume ¢ 1 = ¢2 =: cf> and so we get an
explicit formula for the phase:

</> = 3HT + 2A(I:).

2(K1 + K2)

More general formulas for the phase are computed in Alber, Luther, Marsden and
Robbins [1998b].
These methods of analyzing the three wave interaction also allows one to apply
control ideas to manipulate the phases (as in what are called cascaded nonlineari-
ties); see Alber, Luther, Marsden and Robbins [1998a] for more information.