Lamésche — Mathieusche — und Verwandte Funktionen in Physik und Technik

The spectrum of Hill's equation

“…The following information may be found in Magnus and Winkler [32], see also Strutt [37]. Q, acting on the class of functions f~L2( -c~:), ~) with IIf"H2 < 0o, is self-adjoint with spectrum comprising an infinite number of intervals sn is the customary elliptic function of Jacobi, K being the complete elliptic integral of llaodutus k. 2 C~-is the class of real infinitely differentiable functions of period Z L2 I is defined similarly.…”

The stability of the equilibrium of a pendulum of variable length

“…This is confirmed by a more detailed analytical investigation: by the increase in the degree of approximation and the construction of the Hill determinant. 2,5,7,14,15 This qualitative result was established and proved for the first time by Kochin when solving a similar problem on the parametric oscillations of elastic crankshafts. 2,3 This mechanical effect was also confirmed using a highly accurate numerical-analytical accelerated-convergence method and numerous calculations [17][18][19] (see below and the figure).…”

“…Hence, it is required to construct solutions of two self-adjoint boundary-value problems in eigenvalues and functions with conditions of the first and second kind at the ends of the interval, i.e., to Sturm-Liouville type problems. [14][15][16][17][18][19] Note that the above formulation is non-standard, since the modulation index e is not small and occurs in the coefficients of Eq. (1.5) in a singular form: the coefficients become unbounded for = 0, 1 as e → 1.…”

“…In classical investigations, Ince-Strutt frequency diagrams have been constructed and a detailed mathematical apparatus has been developed. [2][3][4][5][6][7][8]14,15 …”

Stability analysis for multi-parameter linear periodic systems

“…Several books have been written on the subject, and in these several hundred references may be found. The most important books are by [25], [13], [14], [1], [11], [24], [28] and [29].…”