By Demeter Krupka

The ebook is dedicated to contemporary study within the worldwide variational concept on delicate manifolds. Its major aim is an extension of the classical variational calculus on Euclidean areas to (topologically nontrivial) finite-dimensional tender manifolds; to this goal the equipment of worldwide research of differential varieties are used. Emphasis is put on the principles of the speculation of variational functionals on fibered manifolds - suitable geometric buildings for variational ideas in geometry, actual box conception and higher-order fibered mechanics. The publication chapters contain: - foundations of jet bundles and research of differential kinds and vector fields on jet bundles, - the speculation of higher-order necessary variational functionals for sections of a fibred house, the (global) first variational formulation in infinitesimal and imperative kinds- extremal stipulations and the dialogue of Noether symmetries and generalizations,- the inverse difficulties of the calculus of diversifications of Helmholtz kind- variational series conception and its outcomes for the worldwide inverse challenge (cohomology conditions)- examples of variational functionals of mathematical physics. entire formulations and proofs of all simple assertions are given, in response to theorems of worldwide research defined within the Appendix.

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Another consequence of Theorem 1 is concerned with sections of the ﬁbered manifold J r Y over the base X. We say that a section d of J r Y, deﬁned on an open set in X, is holonomic, or integrable, if there exists a section c of Y such that ð37Þ d ¼ J r c: Obviously, if c exists, then applying the projection pr:0 to both sides, we get pr:0 d ¼ c; thus, if c exists, it is unique and is determined by c ¼ pr:0 d: ð38Þ Theorem 2 A section d: U ! ik ¼ 0 for all r, k, and i1 ; i2 ; . ; ik such that 1 1 i1 i2 Á Á Á ik n.

LrÀ1 lr . 40 2 Differential Forms on Jet Prolongations of Fibered Manifolds (b) Consider two charts ðV; wÞ, w ¼ ðxi ; yr Þ, and ðV; wÞ, w ¼ ðxi ; yr Þ, such that V \ V 6¼ [. jm ð27Þ Setting f ¼ ykp1 p2 ÁÁÁpk , where p1 (c) By deﬁnition, p2 ÁÁÁ pk , and using (17), we get (25). jk l J r aÞdðxl J r aÞ: ð28Þ Denote by a0 the p-projection of a. Since from Sect. ip i\i2 \ÁÁÁ\ip À ð30Þ These conditions mean that the section d is of the form d ¼ J r ðpr;0 dÞ as required. 1 The Contact Ideal 41 The basis of 1-forms (24) on V r is usually called the contact basis.

6 Jet Prolongations of Automorphisms of Fibered Manifolds 23 Lemma 8 (a) For any s such that 0 s r, pr J r a ¼ a0 pr ; pr;s J r a ¼ J s a pr;s : ð86Þ (b) If two C r automorphisms a and b of the ﬁbered manifold Y are composable, then J r a and J r b are composable and J r a J r b ¼ J r ða bÞ: Proof All these assertions are easy consequences of deﬁnitions. ð87Þ h Formula (86) shows that J r a is an Cr automorphism of the r-jet prolongation J r Y of the ﬁbered manifold Y, and also Cr automorphisms of J r Y over J s Y.