Lecture 2. Stability, Underwater Vehicle Dynamics and Phases
Some history, background and literature on stability. The energy mo-
mentum method is an extension of the Arnold (or energy-Casimir) method for
the study of stability of relative equilibria for Lie-Poisson systems on duals of Lie
algebras, especially those of fluid dynamical type. The method simultaneously
extends and refines the fundamental stability techniques going back to Routh, Li-
apunov and in more recent times, to Smale [1970].
There are several motivations for developing these extensions. First of all, the
energy-momentum method can deal with Lie-Poisson systems for which there are
not sufficient Casimir functions available for the Arnold method to be effective,
such as 3D ideal flow and certain problems in elasticity. We recall that in the
Arnold method, one seeks a Casimir function C (plus possibly other conservedquantities) such that H + C h as a critical point at the equilibrium in question
and that 82 (H + C) is definite there (or satisfies suitable convexity assumptions).
For 3D Euler flow there is only one known Casimir function, the helicity, which is
not enough to analyze the stability of most equilibria-even at the first variation
step. Abarbanel and Holm [1987] use what we see retrospectively is the energy-
momentum method to show that 3d equilibria for ideal flow are always formally
unstable due to vortex stretching. Other fluid and plasma situations, such as those
considered by Chern and Marsden [1990] for ABC flows, and certain multiple hump
situations in plasma dynamics (see Holm, Marsden, Ratiu and Weinstein [1985] and
Morrison [1987] for example) provided additional motivation for generalizations in
the Lie-Poisson setting.
A second motivation is to extend the method to systems that need not be
Lie-Poisson and still make use of the powerful ideas of reduction (even if reduced
spaces are not literally used), as in the original Arnold method. Examples such as
rigid bodies with vibrating antennas (Sreenath, et al [1988], Oh et al [1989], Krish-
naprasad and Marsden [1987]) and coupled rigid bodies (Patrick [1989]) motivated
the need for such an extension of the theory.
Finally, it gives sharper stability conclusions in material representation and
links with geometric phases, as we shall discuss.
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