TY - JOUR

T1 - Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane

AU - Bialy, M.

AU - Mironov, A. E.

PY - 2019/1/16

Y1 - 2019/1/16

N2 - Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic feld is considered in this paper. The question of the existence of an integral of motion which is a polynomial in the components of the velocity is investigated. It is shown that if such an integral exists, then the boundary of the domain defnes a non-singular algebraic curve in C3. It is also shown that for a domain other than a geodesic disk, magnetic billiards does not admit a polynomial integral for all but perhaps fnitely many values of the magnitude of the magnetic feld. To prove our main theorems a new dynamical system, outer magnetic billiards , on a constant-curvature surface is introduced, a system dual to magnetic billiards. By passing to this dynamical system one can apply methods of algebraic geometry to magnetic billiards.

AB - Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic feld is considered in this paper. The question of the existence of an integral of motion which is a polynomial in the components of the velocity is investigated. It is shown that if such an integral exists, then the boundary of the domain defnes a non-singular algebraic curve in C3. It is also shown that for a domain other than a geodesic disk, magnetic billiards does not admit a polynomial integral for all but perhaps fnitely many values of the magnitude of the magnetic feld. To prove our main theorems a new dynamical system, outer magnetic billiards , on a constant-curvature surface is introduced, a system dual to magnetic billiards. By passing to this dynamical system one can apply methods of algebraic geometry to magnetic billiards.

KW - constant-curvature surfaces

KW - Magnetic billiards

KW - polynomial integrals

KW - CLASSICAL BILLIARDS

KW - magnetic billiards

KW - INTEGRABLE BILLIARDS

KW - SURFACES

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85072679142&origin=resultslist&sort=cp-f&src=s&st1=Polynomial+non-integrability+of+magnetic+billiards+on+the+sphere+and+the+hyperbolic+plane&st2=&sid=78e59cd36babc6d24f0a82a661bf08cf&sot=b&sdt=b&sl=104&s=TITLE-ABS-KEY%28Polynomial+non-integrability+of+magnetic+billiards+on+the+sphere+and+the+hyperbolic+plane%29&relpos=0&citeCnt=0&searchTerm=

U2 - 10.1070/RM9871

DO - 10.1070/RM9871

M3 - Article

AN - SCOPUS:85072679142

VL - 74

SP - 187

EP - 209

JO - Russian Mathematical Surveys

JF - Russian Mathematical Surveys

SN - 0036-0279

IS - 2

M1 - 1

ER -