Towards an algebro-geometric interpretation of the Neumann system
The author considers a point moving on the sphere \(\xi^ 2_ 1+...+\xi^ 2_ n=1\) under the influence of a quadratic potential \(\phi (\xi)=-\sum x_ i\xi^ 2_ i.\) The differential equation describing this motion is \({\ddot \xi}{}_ i+q\xi_ i=x_ i\xi_ i\), \(q=\sum \xi^ 2_ i+x_ i\xi^ 2_ i\), \(\xi^ 2_ 1+...+\xi^ 2_ n=1\) (Neumann's system). This problem was solved via (ultra-elliptic) quadratures by Neumann for \(n=3\). Neumann's system is closely related to the Burchnall-Chaundy-Krichever theory of second order differential operators \(D^ 2+q(t)\). The author derives from this theory Lax equations and constants of motion for Neumann's system.
scientific article; zbMATH DE number 3931606
scientific article