Continuous versions of some discrete maps or what goes on when the lights go out
The author considers the problem of realizing a diffeomorphism \(\phi : M \to M\) as the time-one map for a vector field \(V\) on \(TM\). He describes four examples of discrete maps on manifolds. Each of them is realized as the time-one map for a completely integrable Hamiltonian system. The \(k\)- iterate of the map corresponds to the time \(k\) evaluation for the Hamiltonian system. He states a sort of stroboscope theorem for these examples: what goes on when the lights go out is described by completely integrable Hamiltonian flow. The relevant ingredient from integrable mechanics is the classical \(R\)- matrix. Each example has a corresponding theorem to be found in the literature. Example \(1\) is the \(QR\) algorithm for computing the eigenvalues of a matrix \(M_ 0\). This is a sequence of isospectral matrices that relates to Toda flows [the author, \textit{L. C. Li} and \textit{C. Tomei}, Commun. Pure Appl. Math. 42, No. 4, 443-521 (1989; Zbl 0689.70006)]. Example 2 is the singular value decomposition algorithm for finding the singular values of a matrix \(M_ 0\) (i.e. the spectrum of \(\sqrt {M_ 0^ T M_ 0}\)). The theorem can be found in [the author, \textit{J. Demmel, L. C. Li} and \textit{C. Tomei}, SIAM J. Numer. Anal. 28, No. 5, 1463-1516 (1991; Zbl 0737.65035)]. The 3rd example is an algorithm for computing an elliptic integral by performing successive arithmetic and geometric means on the parameters. For this there is a theorem by the author, Li, Previato, Tomei. The 4th example is a billiard ball problem on an elliptical billiard table considered in [the author, \textit{L. C. Li} and \textit{C. Tomei}, Mem. Am. Math. Soc. 479, 101 p. (1992; Zbl 0770.34005)].
completely integrable Hamiltonian system
scientific article; zbMATH DE number 640334
scientific article