GEOMETRY OF THE KAHAN DISCRETIZATIONS OF PLANAR QUADRATIC HAMILTONIAN SYSTEMS. II. SYSTEMS WITH A LINEAR POISSON TENSOR

摘要

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation" by P. van der Kamp et al. [5], it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form ?(x, y), let B_1,B_2 be any two distinct points on the line ?(x, y) = -c, and let B_3,B_4 be any two distinct points on the line ?(x, y) = c. Set B_0 = 1/2 (B_1+B_3) and B_5 = 1/2 (B_2 + B_4); these points lie on the line ?(x, y) = 0. Finally, let B_∞ be the point at infinity on this line. Let E be the pencil of conics with the base points B1;B2;B3;B4. Then the composition of the B_∞-switch and of the B0-switch on the pencil E is the Kahan discretization of a Hamiltonian vector field f = ?(x, y)(?H/?y-?H/?x)with a quadratic Hamilton function H(x, y). This birational map Φ_f : CP~2 →K CP~2 has three singular points B_0,B_2,B_4, while the inverse map Φ_f~(-1) has three singular points B_1,B_3,B_5.