Local symmetries and conservation laws

Abstract

Abstract. Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of'higher KdV equations' are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomologicai nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that 'symmetry' and 'conservation law' are, in some sense, the 'dual' conceptions which coincides in the 'self-dual' case, namely, for Euler-Lagrange equations. Training examples are also given.

Direct construction method for conservation laws of partial differential equations Part II: General treatment

“…[7,13,11,5,14]) is the explicit delineation of the linear determining system ℓ * ∪ h which incorporates (and identifies) the necessary and sufficient conditions for adjoint symmetries to be multipliers, without moving off the space of solutions of the given PDE(s) G. Consequently, one can calculate multipliers of conservation laws by effective algorithmic procedures. Moreover there is the added computational advantage of allowing the determining equations in the adjoint system ℓ * and the extra system h to be mingled to optimally solve the determining system ℓ * ∪ h, as illustrated by the conservation law classification results for the PDE examples in Part I.…”

A remark on nonlinear accessibility conditions and infinite prolongations

“…We remark here that the strong and weak accessibility Lie rank conditions possess an easy and nice interpretation within the framework of the differential geometry of jets and prolongations of infinite order, which is becoming a mainstay in some parts of mathematics and physics (see, e.g., [3,4,32,38,41,49,66,67,69] and the references therein). Motivated by the study of differential flatness [25], i.e., by dynamic feedback linearizabilty, this infinite-dimensional geometry is now being developed in nonlinear control (cf.…”

“…[26]). We thus introduce the notion of diffiety, i.e., of an abstract infinite-dimensional manifold equipped with a Cartan distribution (see [41,65,66,67,69]), which is justified by an example. We show that an orbitally flat system, which is Lie-Bäcklund isomorphic to a trivial diffiety, does not possess any non-trivial local first integral.…”

“…A local realvalued C ∞ -function on M is defined in a local chart as an usual C ∞ -function of a finite number of coordinates. A diffiety 5 is such a manifold M, which is moreover equipped with an involutive distribution, the Cartan distribution, of finite dimension, called its Cartan dimension (see [41,65,66,67,69] for more details). A local vector field, which is a local section of the Cartan distribution, is called a Cartan field.…”

О Бесконечном Множестве Законов Сохранения Для Бесконечномерных Симметрий

“…РОЗЕНХАУС для любой функции f (u x ) ∈ C 2 выполняются как нётеровы, так и строгие гра-ничные условия (13), (16). Таким образом, уравнение (48) обладает бесконечным множеством существенных законов сохранения D x −au t af (u x ) + 2f (u x ) e aux + D t af (u x ) + 2f (u x ) e aux .…”