Equivalence of discrete euler equations and discrete hamiltonian systems

Abstract

Erbe and Yan recently presented a discrete linear Hamiltonian system. Their system is a special case of the discrete Hamiltonian system Δy(n - l) = Hz(n, y(n), z(n - l))Δz(n - l) = -Hy(n, y(n), z(n - l)), where Δy(n - 1) = y(n) - y(n - 1). Under certain implicit solvability hypotheses, these systems are equivalent to the discrete Euler equation f(hook)y(n, yn, Δyn - l) = Δf(hook)r(n, yn, Δyn - l). A Reid Roundabout Theorem for linear recurrence relations -Knyn+1 + Bnyn - KTn-1yn-1 = 0 is shown to imply the corresponding result obtained by Erbe and Yan for discrete linear Hamiltonian systems. Furthermore, discrete linear Hamiltonian systems are shown to have a symplectic transition matrix. © 1993 Academic Press, Inc.