Cartan’s concept of duality for second-order ordinary differential equations

Crampin, M.; Saunders, D.J.

doi:10.1016/j.geomphys.2004.09.003

Cited by 11 publications

(28 citation statements)

References 8 publications

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“…Finally, for sub-case III.2 we will show that there is essentially one type of configuration of invariant values that gives the rank sequence (̺ 4 ,̺ 5 ,̺ 6 ,̺ 7 ,̺ 8 ,̺ 9 ,̺ 10 ) = (1,4,5,6,7,8,8). We will derive this configuration, and in the subsequent section integrate the corresponding structure equations.…”

Section: Case IIImentioning

confidence: 89%

“…Hence, the "worst-case scenario", as far as the IC order is concerned, is (̺ 5 ,̺ 6 ,̺ 7 ,̺ 8 ,̺ 9 ) = (5,6,7,8,8);…”

Section: The Fundamental Branchingmentioning

confidence: 99%

“…In his study of projective connections, [4], Cartan mentions that there exists a notion of duality among second-order ordinary differential equations. Modern accounts can be found in [5,23], but for completeness, we summarize the construction below. For second-order ordinary differential equations, a solution to an initial-value problem may be represented by a two-parameter family Φ(x, u, x, u) = 0, (A.1)…”

Section: A Cartan's Duality For Second-order Odesmentioning

confidence: 99%

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Point equivalence of second-order ODEs: Maximal invariant classification order

Valiquette

2015

Journal of Symbolic Computation

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We show that the local equivalence problem of second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also demonstrate that, modulo Cartan duality and point transformations, the Painlevé-I equation can be characterized as the simplest second-order ordinary differential equation belonging to the class of equations requiring 10th order jets for their classification. Formalization of the problem and resultsFollowing standard practices, [12,25], we let p = u x , q = u xx denote the first and second order derivatives of a single variable function u = u(x). Then, two second-order ordinary differential equations q = f (x, u, p) and Q = F (X, U, P ), p = u x , q = u xx , P = U X , Q = U XX ,

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Section: Case IIImentioning

confidence: 89%

“…Hence, the "worst-case scenario", as far as the IC order is concerned, is (̺ 5 ,̺ 6 ,̺ 7 ,̺ 8 ,̺ 9 ) = (5,6,7,8,8);…”

Section: The Fundamental Branchingmentioning

confidence: 99%

Section: A Cartan's Duality For Second-order Odesmentioning

confidence: 99%

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Point equivalence of second-order ODEs: Maximal invariant classification order

Valiquette

2015

Journal of Symbolic Computation

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“…A considerable number of works in the literature are devoted to study of problems such as linearization, existence of local coordinates in which an equation takes the special form, and theory of connections associated with differential equations [20][21][22][23][24][25][26][27]. In modern terminology, an ODE or a PDE is identified with an exterior differential system in an appropriate jet bundle and this approach leads to geometrization of differential equations.…”

Section: Introductionmentioning

confidence: 99%

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

Bayrakdar

2018

Advances in Mathematical Physics

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We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers' equations describe surfaces attached to an ODE of the form / = ( , ) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

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“…The 3 × 3 matrix in the upper left hand corner is effectively the normal Cartan projective connection (see [4]), while the bottom right hand corner is its dual (see [3]). The 3 × 3 matrix in the upper left hand corner is effectively the normal Cartan projective connection (see [4]), while the bottom right hand corner is its dual (see [3]).…”

mentioning

confidence: 99%

Fefferman-type metrics and the projective geometry of sprays in two dimensions

Crampin

Saunders

2007

Math. Proc. Camb. Phil. Soc.

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A spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of the sprays, and the geodesics of the sprays determine null geodesics of the metrics. The metrics in question have previously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a different method involving the exploitation of a close analogy between the Cartan geometry of second-order ordinary differential equations and of three-dimensional Cauchy–Riemann structures. From this perspective the derived metrics are seen to be analoguous to those defined by Fefferman in the CR theory, and are therefore said to be of Fefferman type. Our version of the construction reveals that the Fefferman-type metrics are derivable from the scalar triple product, both directly in the flat case (which we discuss in some detail) and by a simple extension in general. There is accordingly in our formulation a very simple expression for a representative metric of the class in suitable coordinates.

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Cartan’s concept of duality for second-order ordinary differential equations

Abstract: We discuss the theory of the second-order ordinary differential equation initiated by Cartan, concentrating especially on Cartan's notion of duality between such equations, and its consequences.

Cited by 11 publications

References 8 publications

Point equivalence of second-order ODEs: Maximal invariant classification order

Point equivalence of second-order ODEs: Maximal invariant classification order

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

Fefferman-type metrics and the projective geometry of sprays in two dimensions

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