Restriction and Dimensional Reduction of Differential Operators (original) (raw)
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Some results on the calculus of variations on jet spaces
Annales De L Institut Henri Poincare-physique Theorique, 1983
The basic object is a fibered manifold p : E -~ M and the framework is that of jet spaces. Given a Lagrangian form A on JE, we work with the space {A} of variational forms associated to A. It is this space which is important in the calculus of variations. We study a new operator (defined only on Ker N c TJE where is the fundamental 1-form) canonically associated to {A }. This operator is well suited for studying critical sections and functorial properties. The so called Euler-Lagrange operator EA appears as an extension of Variational symmetries are introduced as morphisms of a category whose objects are the variational forms { A}. The uniqueness of the Poincare-Cartan form 0A is proved under certain circumstances. Various interesting relations between A, EA and eA are investigated. RESUME. L’objet fondamental est une variete avec un espace fibre p : E ~ M et Ie cadre est celui des espaces de jets. Etant donnée une forme Lagrangienne A sur JE, on considere l’espace {} des formes var...
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In this section we summar~ze the notation and relevant facts of the theory of jet bundles. Our summary is not intended to be complete, but only to explain those parts of the theory which we need later. As we explained in section 1, this theory is useful for the discussion of Backlund transformations because it provides a rigorous basis for the manipulation of partial derivatives as if they were independent variables. The formulation of the theory is due to C. Ehresmann [23]. There are brief mathematical introductions in the book by Golubitsky and Guillemin [35] and in the memoir by Guillemin and Sternberg [35]. More leisurely presentations may be found in several books by R. Hermann [40, 41, 42]. Jet bundles have often been used before in the study of partial differential equations, for example by H.H. Johnson [48],
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In the present paper, which is a sequel of [1], we consider the dimensional reduction of differential operators (DOs) that are invariant with respect to the action of a connected Lie group G. The action of G on vector bundles induces naturally actions of G on their sections and on the DOs between them. In [1] we constructed explicitly the reduced bundle ξ, such that the set of all its sections, C∞(ξG), is in a bijective correspondence with the set C∞(ξ) of all G-invariant sections of the original vector bundle ξ. The main goal of the present paper is, given a G-invariant DO D : C∞(ξ) → C∞(η) to construct the reduced DO D : C∞(ξG) → C∞(ηG). Our construction of D uses the geometrically natural language of jet bundles which best reveals the geometry of the DOs and reduces the manipulations with DOs to simple algebraic operations. Since ξ was constructed in [1] by restricting a certain bundle to a submanifold of its base, an essential ingredient of the dimensional reduction of a DO is t...
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We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise (k, r)-jet. The relations to the Euler-Lagrange morphism of the variational calculus are described.
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The calculus of local variational differential operators introduced by B. L. Voronov, I. V. Tyutin, and Sh. S. Shakhverdiev is studied in the context of jet super space geometry. In a coordinate-free way, we relate these operators to variational multivectors, for which we introduce and compute the variational Poisson and Schouten brackets by means of a unifying algebraic scheme. We give a geometric definition of the algebra of multilocal functionals and prove that local variational differential operators are well defined on this algebra. To achieve this, we obtain some analytical results on the calculus of variations in smooth vector bundles, which may be of independent interest. In addition, our results give a new a new efficient method for finding Hamiltonian structures of differential equations.
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As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals. It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-L...