1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 355

structures found in these papers, we consider the case when P = T*Q for a con-

figuration space Q a symmetry group G acting on Q, with the standard cotangent

bundle momentum map J : T*Q _. g*, where g* is the Lie algebra of G. Thus,

we have the well known formula ( J ( aq), ~) = ( aq, ~Q ( q)). Of course one gets the

Lie-Poisson case when Q = G. In this cotangent bundle case and when H is ki-

netic minus potential, an amazing thing happens: using splittings that are based on

the mechanical connection, the second variation of Ht, at the relative equilibrium

can always be arranged to be block diagonal, while, simultaneously, the symplec-

tic structure also has a simple block structure so that the linearized equations are

put into a useful canonical form. Even in the Lie-Poisson setting, this leads to

situations in which one gets much simpler second variations. This block diagonal

structure is what gives the method its computational power. Roughly speaking, for

rotating systems, this method optimally separates the rotational and vibrational

modes. In fact, links between these methods and problems in molecular dynamics

are most interesting (see Littlejohn and Reinch [1996]).

Lagrangian version of the energy-momentum method. The energy momen-

tum method may also be usefully formulated in the Lagrangian setting and this

setting is very convenient for the calculations in many examples. The general the-

ory for this was done in Lewis [1992b] and Wang and Krishnaprasad [1992]. This

Lagrangian setting is closely related to the general theory of Lagrangian reduction

mentioned in Lecture l. In this context one reduces variational principles rather

than symplectic and Poisson structures and for the case of reducing the tangent

bundle of a Lie group, recall that this leads to the Euler-Poincare equations.

Effectiveness in examples. The energy momentum method has proven its effec-
tiveness in a number of examples. For instance, Lewis and Simo [1990] were able
to deal with the stability problem for pseudo-rigid bodies, which was thought up
to that time to be analytically intractable.
The energy-momentum method can sometimes be used in contexts where the
reduced space is singular or at nongeneric points in the dual of the Lie algebra. This
is done at singular points in Lewis [1992b] and Lewis, Ratiu, Simo and Marsden
[1992] who analyze the heavy top. As we mentioned above, it was extended for
compact groups, to allow nongeneric points μ E g* in Patrick [1992, 1995].

The role of phases. One of the key things in the energy-momentum method is

to keep track of which group drifts are possible. We discuss some basic examples
of phases below. This is very important for the reconstruction process and for
understanding the Hannay-Berry phase in the context of reduction (see Marsden,
Montgomery and Ratiu [1990] and references therein). Noncompact groups come up
in a number of examples , such as the dynamics of rigid bodies in fluids (underwater
vehicles), which we discuss below.
Geometric phases or holonomy is also a useful concept in many locomo-
tion problems as we will explain later. The mathematical foundations for phases in
holonomic systems uses the theory of reduction, both Hamiltonian and Lagrangian;
while for nonholonomic systems this foundation has been laid by Bloch, Krish-
naprasad, Marsden, and Murray [1996], Ostrowski [1996] and many others. For
the study of controll ability and gaits, this setting of Lagrangian reduction has been
very useful. An overview may be found in Lecture 3 and in Marsden and Ostrowski
[1998].