366 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
6(^6 2) 1.5
f
·······
l
........
0........ l
!
........
0.5 0 0.5 1.5
RealFigure 2.8. Eigenvalue movement for relative equilibrium 4 as the vertical
momentum is increased.at ±(IIV I 3 )i is passed by one of the moving eigenvalues. This passing (resonance)
of eigenvalues occurs when Pf is such that
Remarkably, this passing event may be precisely detected by the more delicate
stability criterion (at non-generic points) that was mentioned earlier. In this com-
putation, the inertia and mass matrix parameters were chosen (for illustration only)to be Ii = h = 1, h = 0.5, m1 = m 2 = 1, m3 = 0.8 and m = 0.5, g = 9.8, l = 0.5.
The equilibrium angular momentum is rrg = 2 and Pf is increased from 1 to 6.
The five eigenvalues that remain fixed are drawn as circles (note there are three
eigenvalues at the origin). The crosses indicate the positions of the four remainingeigenvalues at the point when Pf = l. The dotted lines show the paths of these
four eigenvalues as Pf is increased to 6. The eigenvalue crossing occurs when
Pf = J9.8 ~ 3.13 and the Hamiltonian Hopf bifurcation point corresponds to
Pf~ 3.75.It would be interesting to study the Hamiltonian-Krein-Hopf bifurcation in
more detail. (Be warned that standard versions of this theorem will not work.)It would also study the effects of symmetry breaking (such as the S^1 symmetry
of the vehicle in the case of the rising vehicle) needs additional attention. The
techniques of Knobloch, Mahalov and Marsden [1994] may prove useful in this
regard. A complication is the nongeneric nature of the coadjoint orbit, so the theory
of eigenvalue movement (see Dellnitz, Melbourne and Marsden [1992]) requires
additional work.
Some feedback stabilization methods (Bloch, Krishnaprasad, Marsden and San-
chez, [1992]) and Bloch, Leonard and Marsden [1997] may be applied to stabilize
otherwise unstable equilibria. This is an interesting direction for future research;
the basic ideas of the method are given in Lecture 4.