Differential Invariants of Second Order ODEs, I
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Differential Invariants of Second Order ODEs, I Valeriy A. Yumaguzhin
Received: 1 October 2008 / Accepted: 18 December 2008 / Published online: 30 January 2009 © Springer Science+Business Media B.V. 2009
Abstract This paper is devoted to differential invariants of equations y = a 3 (x, y)y 3 + a 2 (x, y)y 2 + a 1 (x, y)y + a 0 (x, y). w.r.t. point transformations. The natural bundle of these equations and its bundles of kjets of sections, k = 0, 1, 2, . . . , are considered. The action of the pseudogroup of all point transformations on these bundles is investigated. Tensor differential invariants distinguishing orbits of this action on jet bundles of second and third orders are constructed. A complete collection of generators and their differential syzygies is obtained for the algebra of all scalar differential invariants of a wide class of considered equations containing generic equations. Keywords 2-nd order ordinary differential equation · Point transformation · Equivalence problem · Differential invariant Mathematics Subject Classification (2000) 53A55 · 53C10 · 53C15 · 34A30 · 34A26 · 34C20 · 58F35 1 Introduction This paper is devoted to differential invariants of ordinary differential equations of the form y = a 3 (x, y)y 3 + a 2 (x, y)y 2 + a 1 (x, y)y + a 0 (x, y).
(1)
There are different approaches to construct differential invariants of these equations, see R. Liouville [15], S. Lie [13, 14], A. Tresse [22, 23], E. Cartan [4], G. Thomsen [21], and R.B. Gardner [6]. In [25] we presented an approach to this problem different from above mentioned ones. This approach is a result of our attempt to transfer a construction of structure functions V.A. Yumaguzhin () Program Systems Institute of RAS, Pereslavl’-Zalesskiy 152020, Russia e-mail: [email protected]
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of a G-structure, see [19, 20], to jet bundles of the natural bundle of corresponding geometric objects. In this paper, we develop and state in detail this approach, see also [27]. Another general approach to construct differential invariants can be found in the works of V. Lychagin and B. Kruglikov [12, 16]. We consider every equation E of form (1) as a geometric structure. To this end, we identify the equation E with the section SE : (x, y) → (x, y, a 0 (x, y), a 1 (x, y), a 2 (x, y), a 3 (x, y)) of the product bundle π : R2 × R4 −→ R2 . Thus the set of all equations (1) is identified with the set of all sections of π . It is well known, see [2], that every point transformation of variables x and y transforms every equation (1) to equation of the same form.1 It follows that every point transformation f of the base of π generates the transformation of sections of π . This means that f is lifted in the natural way to the diffeomorphism f (0) of the total space of π . Thus the bundle π of (1) is a natural bundle and (1) considered as a section of π is a geometric structure, see [1]. Every lifted diffeomorphism f (0) is lifted in the natural way to the diffeomorphism f (k) of the bundle J k π of k-jets of sections
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