2. What are the key words of the text?
3. Give a short summary of the text using the key words. Text 5. History of the Rigid-Body Phase Formula
The history of the rigid-body phase formula is quite interesting and seems to have proceeded independently of the other developments above. The formula has its roots in work of MacCullagh dating back to 1840 and Thomson and Tait [1867, §§123, 126]. (See Zhuravlev [1996] and O'Reilly [1997] for a discussion and extensions.) A special case of formula (1.10.7) is given in Ishlinskii [1952]; see also Ishlinskii [1963]. The formula referred to covers a special case in which only the geometric phase is present. For example, in certain precessional motions in which, up to a certain order in averaging, one can ignore the dynamic phase, and only the geometric phase survives. Even though Ishlinskii found only special cases of the result, he recognized that it is related to the geometric concept of parallel transport. A formula like the one above was found by Goodman and Robinson [1958] in the context of drift in gyroscopes; their proof is based on the Gauss-Bonnet theorem. Another interesting approach to formulas of this sort, also based on averaging and solid angles, is given in Goldreich and Toomre [1969], who applied it to the interesting geophysical problem of polar wander (see also Poincaré [1910]!).
The special case of the above formula for a symmetric free rigid body was given by Hannay [1985] and Anandan [1988, formula (20)]. The proof of the general formula based on the theory of connections and the formula for holonomy in terms of curvature was given by Montgomery [1991a] and Marsden, Montgomery, and Ratiu [1990]. The approach using the Gauss-Bonnet theorem and its relation to the Poinsot construction along with additional results is taken up by Levi [1993]. For applications to general resonance problems (such as the three-wave interaction) and nonlinear optics, see Alber, Luther, Marsden and Robbins [1998].
An analogue of the rigid-body phase formula for the heavy top and the Lagrange top (symmetric heavy top) was given in Marsden, Montgomery, and Ratiu [1990]. Links with vortex filament configurations were given in Fukumoto and Miyajima [1996] and Fukumoto [1997].
1. Ask 5-6 questions on the text.
2. What are the key words of the text?
3. Give a short summary of the text. Text 6. Some History of Poisson Structures
Following from the work of Lagrange and Poisson discussed at the end of §8.1, the general concept of a Poisson manifold should be credited to Sophus Lie in his treatise on transformation groups written around 1880 in the chapter on "function groups." Lie uses the word "group" for both "group" and "algebra." For example, a "function group" should really be translated as "function algebra."
Lie defines what today is called a Poisson structure. The title of Chapter 19 is The Coadjoint Group, which is explicitly identified on page 334. Chapter 17, pages 294-298, defines a linear Poisson structure on the dual of a Lie algebra, today called the Lie-Poisson structure, and "Lie's third theorem" is proved for the set of regular elements. On page 349, together with a remark on page 367, it is shown that the Lie-Poisson structure naturally induces a symplectic structure on each coadjoint orbit. As we shall point out in §11.2, Lie also had many of the ideas of momentum maps. For many years this work appears to have been forgotten.
Because of the above history. Marsden and Weinstein [1983] coined the phrase "Lie-Poisson bracket" for this object, and this terminology is now in common use. However, it is not clear that Lie understood the fact that the Lie-Poisson bracket is obtained by a simple reduction process, namely, that it is induced from the canonical cotangent Poisson bracket on T*G by passing to g* regarded as the quotient T*G/G, as will be explained in Chapter 13. The link between the closedness of the symplectic form and the Jacobi identity is a little harder to trace explicitly; some comments in this direction are given in Souriau [1970], who gives credit to Maxwell.
Lie’s
work starts by taking functions f1,
...
,Fr
on
a symplectic manifold M, with the property that there exist functions
of
r
variables
such that
{Fi,Fj} = Gij(F1,...,Fr).
In
Lie’s time, all functions in sight are implicitly assumed to be
analytic. The collection of all functions
of
f1,
...
,Fr
is the "function group"; it is provided with the bracket
where
and
Considering
F
= (F1,...
,Fr)
as
a map from M to an r-dimensional
space P,
and
and
as functions on P,
one
may formulate this as saying that [
,
]
is
a Poisson structure on P,
with
the property that
.
1. Ask 5-6 questions on the text.