Integrable natural systems in pseudoeuclidean plane.
A Hamiltonian system is called natural if its Hamiltonian is a sum of a positive definite kinetic energy and a potential. Natural Hamiltonian systems have been the object of much attention classically; if a natural Hamiltonian system admits a real-analytic integral, then it admits an integral which is polynomial in the momenta [cf. \textit{E. T. Whittaker}, A treatise on the analytical dynamics of particles and rigid bodies. With an introduction to the problem of three bodies. With a foreword by Sir William McCrea. Reprint of the fourth edition, Cambridge Mathematical Library. Cambridge (UK) etc.: Cambridge University Press (1988; Zbl 0665.70002)]. The authors replace the kinetic energy \({1\over 2}(p_1^2+p_2^2)\) in the 2-dimensional Hamiltonian, with a conic determined by a symmetric \(2\times 2\) matrix \(G\) (the ``pseudoeuclidean'' metric of the title), and classify those systems that admit an additional integral, quadratic in the momenta. By solving for the Darboux-Nijenhuis tensor, the authors give (bi-Hamiltonian) recursion formulas that generate all the polynomial potentials, as well as the solution by quadratures. In the last section, they provide examples related to the celebrated Garnier system, the Hénon-Heiles system, and the two-gap Schrödinger equation (which they deem to be new in the literature).
scientific article; zbMATH DE number 1799977