ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 58F07 Selivanova, Elena N., Universität Tübingen, Germany; Nizhny Novgorod State Pedagogical University, Russia New integrable cases of conservative systems on S^{2} We propose new examples of conservative systems on S^{2} possessing an integral of third and fourth degree in momenta. As particular cases of our construction we obtain the known cases of Goryachev–Chaplygin (degree 3) and Kovalevskaya (degree 4). We prove THEOREM 1. Conservative systems on S^{2} with energy

H = \frac{dφ^{2} + dy^{2}}{ψ^{′2}(y)} − (ψ′′(y) − ψ(y)) ψ^{′2}(y) cos φ

H_{b} = \frac{ψ^{′2}(y) − ψ^{2}(y) + b}{ψ^{′2}(y)} (dφ^{2} + dy^{2}) − \frac{ψ^{′2}(y) (ψ′′(y) − ψ(y))}{ψ^{′2}(y) − ψ^{2}(y) + b} cos φ ,

where ψ is a solution of the initial value problem

x′x′′′ = xx′′−2x^{′′2}+x^{′2}+x^{2} , x(0) = 0 , x′(0) = 1 , x′′(0) = τ , τ ∈ (0, T)

b > b_{∗}(τ) = max_{y∈R}(ψ^{2} − ψ^{′2}) or b < b^{∗}(τ) = min_{y∈R}(ψ^{2} − ψ^{′2})

and T is a certain positive constant, possess a nontrivial integral cubic in momenta. THEOREM 2. Conservative systems on S^{2} with energy

H = A^{− \frac{1}{2}} (dφ^{2} + A^{− \frac{1}{2}} du^{2})−\frac{u}{2} ((a + 2u^{2}) − 2A^{\frac{1}{2}} ) cos φ , −2 < a < 2 , a > 2 ,

H_{p} = \frac{A^{\frac{1}{2}} − u^{2} + p}{A^{\frac{1}{2}}} (dφ^{2} + A^{− \frac{1}{2}} du^{2}) − \frac{u ((a + 2u^{2}) − 2A^{\frac{1}{2}} )}{2(A^{\frac{1}{2}} − u^{2} + p)} cos φ

where A(u, a) = 1 + au^{2} + u^{4} and either a > 2 , p ∈ (−∞, −\frac{a}{2}) ∪ (−1, +∞) or −2 < a < 2 , p ∈ (−∞, −1) ∪ (−\frac{a}{2}, +∞) possess a nontrivial integral of fourth degree in momenta.