Table of contents

In his study of periodic orbits of the three-body problem, Hill obtained a formula connecting the characteristic polynomial of the monodromy matrix of a periodic orbit with the infinite determinant of the Hessian of the action functional. A mathematically rigorous definition of the Hill determinant and a proof of Hill's formula were obtained later by Poincaré. Here two multidimensional generalizations of Hill's formula are given: for discrete Lagrangian systems (symplectic twist maps) and for continuous Lagrangian systems. Additional aspects appearing in the presence of symmetries or reversibility are discussed. Also studied is the change of the Morse index of a periodic trajectory upon reduction of order in a system with symmetries. Applications are given to the problem of stability of periodic orbits.

Bibliography: 34 titles.

In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler-Poisson equations, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known.

Bibliography: 82 titles.

Billiards inside quadrics are considered as integrable dynamical systems with a rich geometric structure. The two-way interaction between the dynamics of billiards and the geometry of pencils of quadrics in an arbitrary dimension is considered. Several well-known classical and modern genus-1 results are generalized to arbitrary dimension and genus, such as: the Poncelet theorem, the Darboux theorem, the Weyr theorem, and the Griffiths-Harris space theorem. A synthetic approach to higher-genera addition theorems is presented.

Bibliography: 77 titles.