On First Integrals of Two-Dimensional Geodesic Flows

Abstract

This paper is devoted to searching for Riemannian metrics on 2surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are constructed. Few superintegrable systems are found having both a polynomial and a rational integrals which are functionally independent of the Hamiltonian.

New examples of non-polynomial integrals of two-dimensional geodesic flows *

“…Multiplying the relation {F, H} = 0 by (f 2 p 1 + g 2 p 2 ) 1−m we obtain a homogeneous cubic polynomial in p 1 , p 2 . As in section 3, we have (see [2,25] for details)…”

“…Non-polynomial first integrals of geodesic flows as well as of other Hamiltonian systems are also of interest [17][18][19][20][21][22][23][24][25][26][27]. In general case the problem of searching and classification of such integrals is obviously more complicated.…”

Integrable magnetic geodesic flows on 2-surfaces *

“…Suppose that the magnetic geodesic flow (1.1) admits a linear integral F = a(x, y)p 1 + b(x, y)p 2 + c(x, y) at a fixed energy level {H = C 1 } or, equivalently (e.g. see [33]), at all energy levels. The condition {F, H} mg ≡ 0 implies:…”

“…At the same time a large number of papers are devoted to study rational integrals of mechanical systems at large (including the standard geodesic flows without magnetic fields), e.g. see [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”

Integrable magnetic geodesic flows on 2-surfaces

“…2Λ(x,y) . Suppose that the magnetic geodesic flow (1.1) admits a linear integral F = a(x, y)p 1 +b(x, y)p 2 +c(x, y) at a fixed energy level {H = C 1 } or, equivalently (e.g., see [33]), at all energy levels. The condition {F, H} mg ≡ 0 implies:…”