397 research outputs found
Separation of Variables and the Geometry of Jacobians
This survey examines separation of variables for algebraically integrable
Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these
cases there is a natural class of systems which admit separations in a nice
geometric sense. This class includes many of the well-known cases.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
The Calogero-Fran\c{c}oise integrable system: algebraic geometry, Higgs fields, and the inverse problem
We review the Calogero-Fran\c{c}oise integrable system, which is a
generalization of the Camassa-Holm system. We express solutions as (twisted)
Higgs bundles, in the sense of Hitchin, over the projective line. We use this
point of view to (a) establish a general answer to the question of
linearization of isospectral flow and (b) demonstrate, in the case of two
particles, the dynamical meaning of the theta divisor of the spectral curve in
terms of mechanical collisions. Lastly, we outline the solution to the inverse
problem for CF flows using Stieltjes' continued fractions.Comment: 22 pages, 2 figure
- …