356 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRYIn the underwater vehicle problem, the phase drifts one gets are interesting and
important. They can be understood using the general machinery of reduction by
stages. We return to this topic later in this lecture.
A related problem is the orbit transfer problem in which one tries to find
a control to make a transition from one relative equilibrium to another using the
natural dynamics as much as possible-for example, homoclinic connections provide
dynamical channels. This is how some very interesting trajectories are generated
for spacecraft mission planning.
Usefulness for pde. Arnold [1966c] was the first to prove a nonlinear stability
theorem (the nonlinear Rayleigh inflection point criterion) for the two dimensional
Euler equations of an ideal fluid, so these techniques are clearly effective. For
an exposition of this and many related references, see Arnold and Khesin [1997].
Many other pde problems have b een done as well, cf. Holm, Marsden, Ratiu and
Weinstein [1985] for instance, as well as Lewis [1989]. The Benjamin-Bona theorem
(Benjamin [1972], Bona [1975]) on stability of solitons for the KdV equation can
be viewed as an instance of the energy momentum method, including all the pde
technicalities. See also Maddocks and Sachs [1993], and for example, Oh [1987],
Grillakis, Shatah and Strauss [1987], although of course there are many subtleties
special to the pde context.
Hamiltonian bifurcations. The energy-momentum method has also been used
in the context of Hamiltonian bifurcation problems. One such context is that of
free boundary problems, building on the work of Lewis, Montgomery, Marsden
and Ratiu [1986] which gives a Hamiltonian structure for dynamic free boundary
problems (surface waves, liquid drops, etc), generalizing Hamiltonian structures
found by Zakharov. Along with the Arnold method itself, this is used for a study of
the bifurcations of such problems in Lewis, Marsden and Ratiu [1987], Lewis, [1989,
1992a], Kruse, Marsden, and Scheurle [1993] and other references cited therein.
Converse to the energy-momentum method. Because of the block structure
mentioned, it has also been possible to prove a "converse" of the energy-momentum
method. The idea is to show that, if the second variation is indefinite, then the
system is, in a sense we shall explain, unstable. One cannot, of course hope to
do this literally since there are many systems ( eg, examples studied by Chetayev)
which are formally unstable, and yet their linearizations have eigenvalues lying on
the imaginary axis. Most of these are presumably unstable, but this is a very
delicate situation to prove analytically. Instead, the converse shows something
easier but probably more important: with the addition of dissipation, the system
is destabilized. This idea of dissipation induced instability goes back to Thomson
and Tait in the last century. In the context of the energy-momentum method,
Bloch, Krishnaprasad, Marsden and Ratiu [1994,1996] show that with the addition
of appropriate dissipation, the indefiniteness of the second variation is sufficient to
induce linear instability in the problem.
There are related eigenvalue movement formulas (going back to Krein) that
are used to study non-Hamiltonian perturbations of Hamiltonian normal forms in
Kirk, Marsden and Silber [1996]. There are interesting analogs of this for reversible
systems in O 'Reilly, Malhotra and Namamchchivaya [1996]. These works contain
citations to many more interesting references on these problems.