1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 365

with the addition of a buoyancy potential:

where we determine A , B , C by inverting the block matrix giving the metric (or

mass matrix):

A (J - DM-^1 D T)-^1

B -CDT J-^1 = -M-^1 D T A

c (M - D T J -^1 D)-^1.

The Lie-Poisson equations of motion are

tr IT x n + P x v - mgf x re

:P P x n

t r x n.

R ecall that a point O' = (IT, P , f) E tu * 'is called generic if the coadjoint

orbit through that point has maximal dimension six. These are three independent

Casimir functions (functions invariant under the coadjoint action):

A point O' = (IT, P , f) E tu * is nonge neric if the vectors P and fare not parallel.

If P and r are not both zero, the coadjoint orbit through O' h as dimension four.

Besides t he three Casimirs defined a bove, two additional co nserved quantities

(called subcasimirs) on the nongeneric coadjoint orbits are

C4(IT, P,r) =IT·P, Cs(IT,P,r) =IT·f.

If P = r = 0 with IT "I- 0, then t he coadjoint orbit through w has dimension two.

An additional subcasimir on this nongeneric coadjoint orbit is

Continuing along these lines with the energy-momentum and energy-Casimir

method gives the stability condit ions as stated earlier; we refer to Leonard [1997]

and Leonard and Marsden [1997] for details.

Hamiltonian Hopf bifurcation. Consider now t he fourth relative equilibrium.

The eigenvalues of the linearization of the dynamics at the equilibrium as the equi-

librium linear momentum Pf is varied, are shown in Figure 2.8.

There are three eige nvalues fixed at the origin and two eigenvalues fixed at

±(ITV fJ)i. The remaining four eigenvalues move as the parameter Pf is varied.

These eigenvalues are on the imaginary axis as long as t he stability condition is

met. Assuming m 3 < m 2 (i.e., l3 > l2) and mgl > 0, as Pf is increased , the

pair of eigenvalues above the real axis and the pair below each meet and then split

off t he imagina ry axis-saddle points develop. The point at which each pair of

eige nvalues meets, i.e., t he Hamiltonian-Krein-Hopf bifurcation point, corresponds

to the value of Pf that makes the stability condition an equ ality. Thus, the stability

analysis accurately predicts the bifurcation point. Before t he bifurcation occurs,

i.e., while the eigenvalues are all on the imaginary axis, each of the eigenvalues fixed