1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 359

This motion is stable modulo translations in any direction provided that the system

is bottom heavy (l > 0) and

m2 > m1, mgl > (-

1

• 
  
  
  
  • _2) (P~)^2.
    m2 m3
  
  
  
  

center of buoyancy

g ~

::-------~?~

-r-~

center of gravity

Figure 2.3. Adding buoyancy.

Relative equilibrium 3. The third example has a nongeneric value of the mo-

mentum and is the same as Case 1 except that there is no translation; i.e., P~ = 0.

This motion is stable modulo rotations about the third axis and translations

in any direction provided that h is largest or smallest of the moments of inertia.

(Unlike the free rigid body, this need not mean stability of rotations about the

major or minor axis of the body.)

Relative equilibrium 4. This case is a lso nongeneric but for a different reason;

now the direction of translation is parallel to direction of gravity. Here,

• the centers of buoyancy and gravity are noncoincident and are oriented with
  t he third principal axis parallel to the direction of gravity


• the vehicle is symmetric about the third axis


• the vehicle is rising (or falling) with momentum P~ along the third axis
  rotates with angular momentum rrg. See Figure 2.4.
  This motion is stable modulo rotations about the third axis and translations in
  any direction provided that

(

mgl > -^1 - -1 ) (P^0 2 ii2^0 2

m3 m2^3 ) - -(II 4 3 ) ,
where
_ m1
a2 =.
m1h - m^2 l2
There is an interesting bifurcation in relative equilibrium 4: a Hamiltonian
Krein-Hopf bifurcation (the splitting of eigenvalues off the imaginary axis) and the
passing (resonance) of eigenvalues. We will return to this shortly.
Stability theorem for nongeneric momenta. Now we give a few of the details
on the method and how it works. The setting is a phase space P (a symplectic
manifold) with symplectic form D. Assume as before, that we have a Lie group
G with a Lie algebra g and that G acts symplectically on P with an equivariant