Ueber arithmetische Eigenschaften analytischer Functionen

Abstract

die ~unkte einer beliebigen a b z ~ h l b a r e n _Punktmenge im Gebiete der unbeschrgnkt ver~inderlichen complexen G-r6sse x be~eichnen, wghrend Q irgend eine in dieser •bene i~b e r a l l d i c h t e Tunktmenge bedeuten sell. Dann giebt es stets unendlivh viele e i n d e u t i g e a n a t y t i s c h e t P u n c t i o n e n f (s) , die fi~r aT, le Argumente xo, xi, 9 9 9 tier Menge t ~ nut Werthe aus der Menge Q annehmen. B e w eis. Man bilde die ganzen ralionalen Funetionen: 9 o(X) = 1 , ~t (x)-= s-So, ~(x) = (X-S o) (z-x ,) ,

Hidden symmetries of dynamics in classical and quantum physics

“…Stäckel (Stäckel, 1893) showed that in the case of orthogonal coordinates the geodesic Hamilton-Jacobi is separable if and only if…”

Approximate integrals of motion

“…Gravitational potentials with explicitly known second integrals of motion are rare, the most frequently used in astronomy are axisymmetric potentials and also Stäckel (1893) potentials 1 that cover a large class of potentials, but are not always sufficiently realistic. Such explicit forms are useful, for instance, to understand more clearly the underlying physics of dynamical systems, to build stationary distribution functions, etc.…”

Remarks on the connection between the additive separation of the Hamilton–Jacobi equation and the multiplicative separation of the Schrödinger equation. I. The completeness and Robertson conditions

“…With a smooth real function V ͑potential energy͒ on a Riemannian manifold (Q n ,g) ͑configuration manifold͒ we associate two differential equations, the time-independent Hamilton-Jacobi equation 1 2 ٌW•ٌWϩVϭE, ͑1.1͒…”

“…[1][2][3] Theorem 1.1: ͑Stäckel, 1893͒ The Hamilton-Jacobi equation is separable in orthogonal coordinates q គ if and only if the diagonal components g ii of the metric tensor and the potential V have the form It happens that for solutions of this kind, Eqs.…”