By A. M. Vinogradov

This publication is devoted to basics of a brand new idea, that's an analog of affine algebraic geometry for (nonlinear) partial differential equations. This conception grew up from the classical geometry of PDE's originated by means of S. Lie and his fans by means of incorporating a few nonclassical principles from the idea of integrable platforms, the formal conception of PDE's in its sleek cohomological shape given by means of D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). the most results of this synthesis is Secondary Calculus on diffieties, new geometrical gadgets that are analogs of algebraic forms within the context of (nonlinear) PDE's. Secondary Calculus unusually finds a deep cohomological nature of the final idea of PDE's and shows new instructions of its additional development. contemporary advancements in quantum box idea confirmed Secondary Calculus to be its usual language, promising a nonperturbative formula of the idea. as well as PDE's themselves, the writer describes latest and power functions of Secondary Calculus starting from algebraic geometry to box idea, classical and quantum, together with components similar to attribute periods, differential invariants, concept of geometric constructions, variational calculus, keep watch over concept, and so forth. This publication, concentrated more often than not on theoretical facets, varieties a common dipole with Symmetries and Conservation legislation for Differential Equations of Mathematical Physics, quantity 182 during this related sequence, Translations of Mathematical Monographs, and exhibits the idea ""in action"".

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E. e. in [t 1,t2]. ), i = 1, ... , n. ). 2 Statement of Necessary Conditions This will be referred to as the Euler necessary condition, or as the Euler equations (E i ), i = 1, ... , n. 4 for a proof. e. in [0,1] with an AC function, and with this identification the Euler equation (E 1 ) is (d/dt)(2x') = 2x, or x" = x. The optimal solutions, if any, are arcs of x(t) = C 1 cosh t + Cz sinh t. 2, for more examples. We may not repeat every time the need for the identification mentioned above (Cf. i, ii, iii)).

Let us consider a problem of minimum, for instance. First, the class Qo is not empty, since Yo E Qo. Y. ,/3). Then, y E Q, and I[y] = Ir a 11 + lOPa + l12p fo(t,y(t), y'(t))dt = I[ XO] + (I[y] - I[yo]) < I[ X O], a contradiction. Y. = tb /3 = t 2, X is optimal also in the (restricted) class of all elements y of Q with y(td = x(td, y(t 2) = X(t2)' D E. Strong and Weak Local Minima and Maxima: The General Case In problems of the calculus of variations we may be led in a natural way to consider classes Q of AC functions x(t), t1 ~ t ~ t 2, defined in different intervals.

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