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Text 2. Nonlinear Stability



 

There are various meanings that can be given to the word "stability." In­tuitively, stability means that small disturbances do not grow large as time passes. Being more precise about this notion is not just capricious math­ematical nitpicking; indeed, different interpretations of the word stability can lead to different stability criteria. Examples like the double spherical pendulum and stratified shear flows, which are sometimes used to model oceanographic phenomena show that one can get different criteria if one uses linearized or nonlinear analyses (see Marsden and Scheurle [1993a] and Abarbanel, Holm, Marsden, and Ratiu [1986]).

Some History.The history of stability theory in mechanics is very com­plex, but certainly has its roots in the work of Riemann [1860, 1861], Routh [1877], Thomson and Tait [1879], Poincaré [1885, 1892], and Liapunov [1892, 1897].

Since these early references, the literature has become too vast to even survey roughly. We do mention, however, that a guide to the large Soviet literature may be found in Mikhailov and Parton [1990].

The basis of the nonlinear stability method discussed below was origi­nally given by Arnold [1965b, 1966b] and applied to two-dimensional ideal fluid flow, substantially augmenting the pioneering work of Rayleigh [1880]. Related methods were also found in the plasma physics literature, notably by Newcomb [1958], Fowler [1963], and Rosenbluth [1964]. However, these works did not provide a general setting or key convexity estimates needed to deal with the nonlinear nature of the problem. In retrospect, we may view other stability results, such as the stability of solitons in the Korteweg–de Vries (KdV) equation (Benjamin [1972] and Bona [1975]) as being instances of the same method used by Arnold. A crucial part of the method exploits the fact that the basic equations of nondissipative fluid and plasma dynam­ics are Hamiltonian in character. We shall explain below how the Hamiltonian structures discussed in the previous sections are used in the stability analysis.

 

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