Equivalence of discrete Euler equations and discrete Hamiltonian systems (Q1323120)

The following results are proved: (i) Under certain solvability hypotheses, discrete Hamiltonian systems of the form \[ \Delta y (n-1) = H_ z (n, y(n), z(n-1)),\;\Delta z (n-1) = H_ y (n, y (n), z (n- 1))\tag{1} \] are equivalent to the discrete Euler equation \(f_ y (n, y_ n, \Delta y_{n-1})=\Delta f_ r (n, y_ n, \Delta y_{n-1})\) (here \(r\) denotes the third variable of \(f\)). (ii) A Reid Roundabout Theorem for linear recurrence relations implies the corresponding result obtained by \textit{L. H. Erbe} and \textit{P. Yan} [ibid. 167, No. 2, 355-367 (1992; Zbl 0762.39003)] for discrete linear Hamiltonian systems. These systems are special cases of (1). (iii) Discrete linear Hamiltonian systems have a symplectic transition matrix.