Topics in Modern Mathematics: Petrovskii Seminar

ISBN-13: 9781468416558 / Angielski / Miękka / 2012 / 342 str.

O. Oleinik

Opis

1.1. Nearly Integrable Hamiltonian Systems. In this work we examine the system of Hamiltonian equations i = _ iJH, = iJH iJcp iJl with the Hamiltonian function H = Ho(l) ] eH. (I. cp). (1.1) where E: 1 is a small parameter, the perturbation E: Hl (I, cp) is 2n- periodic in CP=CP1,"'CPS' and I is an s-dimensional vector, I = Il, --- I s The CPi are called angular variables, and the Ii action variables. A system with a Hamiltonian depending only on the action variables is said to be integrable, and a system with Hamiltonian (1.1) is said to be nearly integrable. The system (1.1) is also called a perturbation of the system with Hamiltonian Ho. The latter system is called un- perturbed. 1.2. An Exponential Estimate of the Time of Stability for the Action Variables. Let I(t), cp(t) be an arbitrary solution of the per- turbed system. We estimate the time interval during which the value I(t) differs slightly from the initial value: II(t)-I(O) I 1. The main result of the work is Theorem 4.4 (the main theorem) which is proved in 1]. This theorem asserts that the above-mentioned interval is estimated by a quantity which grows exponentially as the value of perturbation decreases linearly: 1/(t)-/(O)I 0 and b > 0 are given l.n Sec. 4 IJ.

Spis treści

An Exponential Estimate of the Time of Stability of Nearly Integrable Hamiltonian Systems. II.- Local Orbitals of Normal Forms of Vector Fields on a Plane.- Neighborhoods of Zero Type in Embedded Complex Tori.- Solution of a Hyperbolic Cauchy Problem with Characteristic Points in the Initial Surface.- A Variant of a Phragmén-Lindelöf Theorem for Elliptic Equations with Coefficients that are Periodic Functions of all Variables Except One.- Qualitative Theory of Open Homogeneous Cosmological Models with Motion of Matter.- Versal Deformations of Equivariant Vector Fields for the Cases of Symmetries of Order 2 and 3.- On Schreier Varieties of n-Groups and n-Semigroups.- Two-Variable Identities in the Lie Algebra sl(2, k).- Some Factor Groups of a Free Product.- Quasilinear Parabolic Equations and Systems with Two Independent Variables.