1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 369

area=A

Figure 2.11. A parallel movement of a vector around a spherical triangle

produces a phase shift.

vertical direction horizontal directions

bundle

A--: __ J~spaoe

Figure 2.12. A connection divides the space into vertical and horizontal directions.

We now want to reconstruct the dynamics of the original system on C^3 from that

on the three wave surface.

A standard direct approach would be to proceed as follows. First, use (two of)

the constants of motion K 1 , K2, K 3 along with (one of) the reduced coordinates Z 1

and Z2 to determine the amplitudes lq11^2 , lq2 l^2 , lq 3 l^3. Then, write qk = jqk I exp( i¢k)

and substitute into the equations of motion; this yields evolution equations for the
phases ¢k which may then be integrated. While the amplitudes undergo periodic
orbits for most initial data on the reduced phase space (other than the homoclinic
orbits), the phases associated with them obtain a shift, which is then implicitly
obtained via quadratures.