Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard
type: Item
instance of: scholarly article
title: Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard
zbMATH Open document ID: 1263.37059
DOI: 10.1134/S0081543811040092
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2011-10-25T00:00:00Z
review text: The author studies the stability of periodic 2-orbits of the billiard dynamics on a higher dimensional convex table \(\mathcal{D}\) and provides several viewpoints from mechanics (variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points, caustics). Note that the caustics of a billiard table \(\mathcal{D}\) consists of the focal points of \(\Gamma=\partial\mathcal{D}\) with respect to points \(z\in\Gamma\). Moreover, the author gives an estimate of the dimension of the unstable subspace at these orbits in terms of the geometry of the caustic and the Morse indices of the length function. More precisely, let \(z\) be a point on the boundary \(\Gamma\) and \(I(z)\) be the segment on \(\mathcal{D}\) intersecting \(\Gamma\) orthogonally at \(z\). Denote by \(L(z)\) the length of the segment \(I(z)\). Then Theorem 1 states that, under some mild conditions, the dimension of the unstable subspace at a periodic 2-orbit equals to the Morse index. Theorem 2 calculates the characteristic polynomial of the Poincaré operator as the determinant of a matrix of only half the dimension. Hill's formula is also generalized to the higher dimensional case. Sufficient conditions are given to ensure that all multipliers lie on the unit circle.
Mathematics Subject Classification ID: 37D50
zbMATH DE Number: 5963534
zbMATH Keywords: multidimensional billiard
zbMATH Keywords: two-link trajectory
zbMATH Keywords: length function
zbMATH Keywords: caustics
zbMATH Keywords: focal points
zbMATH Keywords: Morse index
zbMATH Keywords: Hill's formula
author: Valery V. Kozlov
reviewed by: Hongkun Zhang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5963534