Automorphisms of Ordinary Differential Equations

2016

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Abstract. The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher-order infinitesimal symmetries which provide a simplifying "linear aproximation" of general groups of higher-order symmetries. The classical Lie's approach is appropriately adapted. PrefaceIf the invertible higher-order transformations of differential equations are accepted as a reasonable subject, the common Lie-Cartan's methods are insufficient for complete solution of the symmetry problem. We recall that even the structure of all higher-order symmetries of the trivial (empty) systems of differential equations (that is, of the infinite-order jet spaces without any differential constraits) is unknown [1,2,3]. The same can be said for the "linearized theory" of the higher-order infinitesimal transformations treated in this article.Let us outline the core of the subject. We start with surfaceslying in the space R m+n with coordinates x 1 , . . . , x n , w 1 , . . . , w m . The higherorder transformations are defined by formulaē, ··) (i, i ′ = 1, . . . , n; j, j ′ = 1, . . . , m) (1.1)2010 Mathematics Subject Classification. 54A17, 58J99, 35A30.

Section: Prefacementioning

confidence: 99%

Section: Fundamental Conceptsmentioning

confidence: 99%

“….) of diffiety Ω such that L D π r = π r+1 , the standard basis [4]. Then Z is a variation if and only if π r+1 (Z) = Dπ r (Z) and therefore the formula…”

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

“…If there is only one string, it does not change after any symmetry [4,Theorem 26]. We have omitted some "noncontrollability" submodules R ⊂ Ω of a finite dimension here.…”

Section: Further Perspectivesmentioning

confidence: 99%

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On the internal approach to differential equations 3. Infinitesimal symmetries

2016

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On the internal approach to differential equations. 1. The involutiveness and standard basis

2016

The article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are subject to arbitrary transformations of variables in the widest possible sense. In this preparatory Part 1, the involutivity and the related standard bases are investigated as a technical tool within the framework of commutative algebra. The particular case of ordinary differential equations is briefly mentioned in order to demonstrate the strength of this approach in the study of the structure, symmetries and constrained variational integrals under the simplifying condition of one independent variable. In full generality, these topics will be investigated in subsequent Parts of this article.

The Poincaré-Cartan forms of one-dimensional variational integrals

2020

Fundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.