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 ƒy(n, yn, Δyn − l) = Δƒr(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.