Content uploaded by Holger R. Dullin
Author content
All content in this area was uploaded by Holger R. Dullin on Dec 18, 2016
Content may be subject to copyright.
A preview of the PDF is not available
Der Kowalewskaja-Kreisel. The Kovalevskaya Top.
Abstract and Figures
This is the booklet accompanying a film on the Kovalevskaya Top, Film C 1961 produced in collaboration with the Institut für den Wissenschaftlichen Film, Göttingen. The movie is available at
http://av.tib.eu/media/10361
Figures - uploaded by Holger R. Dullin
Author content
All figure content in this area was uploaded by Holger R. Dullin
Content may be subject to copyright.
ResearchGate has not been able to resolve any citations for this publication.
Integrable systems which do not have an “obvious” group symmetry, beginning with the results of Poincaré und Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960’s. Although a number of fundamental methods of mathematical physics were based essentially on the perturbation-theory analysis of the simplest integrable examples, ideas about the structure of nontrivial integrable systems did not exert any real influence on the development of physics.
In recent years, the Kowalewski top has received a lot of attention, not only because of the ingenious way Sophie Kowalewskaya integrates her system, but also because of its hidden symmetries. Exactly one hundred years ago, Kowalewski [14], [15] found that not only are the Euler and Lagrange rigid body motions integrable in terms of Abelian integrals, but also a third solid body motion-named after her. In her famous Acta Mathematica paper, she integrates the problem in terms of hyperelliptic integrals, using a very beautiful but mysterious change of variables. When some of us looked at that problem we amved at what seemed like a different conclusion. Namely, we found that the invariant surfaces could be completed, via the flow, into complex algebraic tori (Abelian surfaces) T 2 = Cz/A, where the lattice A is spanned by the columns of the period matrix. 0 2 b c with positive definite imaginary 2. That is to say the problem is not expressed in terms of hyperelliptic integrals (because of the distinct integers 1 and 2 appearing in the matrix), but rather in terms of other kinds of Abelian integrals; see Lesfari [16], [17] and Adler-van Moerbeke [3], [4]. Such Abelian surfaces come up naturally as Prym varieties of double covers of elliptic curves ramified over four points; see Haine [13] and Barth [61, [71.' The paradox alluded to above is well known in algebraic geometry. On the one hand taking half the second period in (1) turns the Abelian surface T 2 into a new Abelian surface J of which TZ is a double covering, as discussed in Section 5. J is principally polarized, as the jargon calls it, and is thus the Jacobian of a 'In fact, W : Bath has pointed out that Abelian surfaces with period matrix (11, which are embeddable into P', are easier to study than the customary principally polarized surfaces, which can at best be embedded into Ps.
This text grew out of graduate level courses in mathematics, engineering and physics given at several universities. The courses took students who had some background in differential equations and lead them through a systematic grounding in the theory of Hamiltonian mechanics from a dynamical systems point of view. Topics covered include a detailed discussion of linear Hamiltonian systems, an introduction to variational calculus and the Maslov index, the basics of the symplectic group, an introduction to reduction, applications of Poincaré's continuation to periodic solutions, the use of normal forms, applications of fixed point theorems and KAM theory. There is a special chapter devoted to finding symmetric periodic solutions by calculus of variations methods.
The main examples treated in this text are the N-body problem and various specialized problems like the restricted three-body problem. The theory of the N-body problem is used to illustrate the general theory. Some of the topics covered are the classical integrals and reduction, central configurations, the existence of periodic solutions by continuation and variational methods, stability and instability of the Lagrange triangular point.
Ken Meyer is an emeritus professor at the University of Cincinnati, Glen Hall is an associate professor at Boston University, and Dan Offin is a professor at Queen's University.
The structure of the integral manifolds of the problem concerning the motion of a heavy rigid body about a fixed point is analyzed for the Kovalevskaya case. An analytical description is obtained for the bifurcation set, and bifurcation diagrams are plotted. For each connected component of the complement of the bifurcation set, in the space of first integral constants, the number of two-dimensional tori appearing in the integral manifold is determined.
This paper is based on the series of lectures which were delivered by the author in 1989 at the Mathematical Sciences Research Institute in Berkeley. As part of the year-long 1988–1989 program in Symplectic Geometry and Mechanics, the Berkeley Mathematical Sciences Research Institute hosted a two-week workshop on the Geometry of Hamiltonian Systems on the period June 5 to June 16, 1989.
