Application of the theory of separability of second-order differential equations

Abstract

A comprehensive application is presented of a recent theory concerning a geometric characterization of separable second-order differential equations. The main purpose of the paper is to illustrate how the practical algorithm developed from this theory effectively works, and what the significance is of the different conditions entering the separability theorem. These conditions are recalled in a coordinate representation, so as to make the paper sufficiently self-contained for practical computations.

A Case Study of Separability of Second-Order Differential Equations

“…Summarizing so far, the requirements R = 0 and C V Φ = 0 impose the three conditions (28, 29, 34). One would expect that the remaining requirement [ ∇ Φ, Φ] = 0 brings more of such complicated computations, but quite surprisingly, in sharp contrast with the situation for the f -system, we will show that this time the complicated first two of the requirements (15) imply the third! Lemma 1.…”

“…Recall first, as we already concluded towards the end of Section 3 for arbitrary dimension, that when A = 0 the three conditions (15) for separability reduce to the same b-conditions for both the f -system and the f -system, despite the fact that they have a different origin, namely a vanishing curvature condition in case of the f -system, and the vanishing [∇Φ, Φ] condition in case of the f -system. But diagonalizability of Φ remains an issue then for the f -system, while Φ = 0 and actually also t = 0 when it concerns the f -system, so that separability should always work there.…”

“…3.2 General separability conditions for the f -system Despite a somewhat similar look, and the resulting observation we just made, the computation of the separability conditions (15) for a sode of the form (2) are much more involved! The defining relation for the Jacobi endomorphism reduces to Φi…”

“…In conclusion, what we have learned about the important conditions (15) for separability in the main theorem is that they translate into (21, 22, 23) for the f -system and into (28, 29, 34) for the f -system. Note that when A = 0 both sets reduce to, what we will call further on the b-conditions ((21) or ( 28)).…”

“…Concerning applications of the geometrical theory, there has never been a systematic study of the separability conditions for specific classes of differential equations. Only occasional illustrative examples have been presented (plus a thorough test case study of an artificially set-up 3-dimensional sode with no less than 18 real parameters, see [15]). The purpose of the present paper is precisely to venture such a more systematic study by picking out a reasonably simple looking non-linear class of second-order differential equations, which preferably should have some geometrical and/or physical relevance and may lead to interesting side issues, inspired by the above mentioned recent developments.…”