Ampere

Abstract: Abstract Three quarters of a century elapsed between Ampère's definition of electrodynamics and Einstein's reform of the concepts of space and time. The two events occurred in utterly different worlds: the French Academy of Sciences of the 1820s seems very remote from the Bern patent office of the early 1900s, and the forces between two electric currents quite foreign to the optical synchronization of clocks. Yet Ampère's electrodynamics and Einstein's relativity are firmly connected through an historical chain involving German extensions of Ampère's work, competition with British field conceptions, Dutch synthesis, and fin de siècle criticism of the aether-matter connection. Darrigol's book retraces this intriguing evolution, with a physicist's attention to conceptual and instrumental developments, and with an historian's awareness of their cultural and material embeddings. This book exploits a wide range of sources, and incorporates the many important insights of other scholars. Thorough accounts are given of crucial episodes such as Faraday's redefinition of charge and current, the genesis of Maxwell's field equations, or Hertz' experiments on fast electric oscillations. Thus emerges a vivid picture of the intellectual and instrumental variety of nineteenth century physics. The most influential investigators worked at the crossroads between different disciplines and traditions: they did not separate theory from experiment, they frequently drew on competing traditions, and their scientific interests extended beyond physics into chemistry, mathematics, physiology, and other areas. By bringing out these important features, this book offers a tightly connected and yet sharply contrasted view of early electrodynamics.

Abstract: A classical problem of a local classification of non-linear equation arising in S. Lie works is studied for the most natural class of Monge-Ampere equations (M.A.E.) on a smooth manifold M\". We solve this problem for a generic classical (n = 2) case and give full proofs of S. Lie classical classification theorems. For multidimensional generalizations of M.A.E. we reduce the problem to a problem in invariant theory which we solve for n=3 and give a partial description for n^4. C-classification is obtained for n=3. Our approach is based on the deep relations between M.A.E. on M\" and contact geometry of J M\"-the first jets space of M\". This approach provides a possibility to apply symplectic and contact geometry methods in classical invariant theory and for calculation Spencer cohomologies as well. In 1B74 Sophus Lie raised the following problem: find the classes of local equivalence of non-linear 2nd order differential equations with respect to the group of contact diffeomorphisms. He formulated theorems on reducing the Monge-Ampere type differential equations to quasilinear and linear form [11]. As far as we know, a complete proof of this theorem had never been published. The class of Monge-Ampere equations is a natural setting for a classification activity since the problem of local classification of arbitrary non-linear 2nd order equations contains as a subproblem the classification with respect to fractionally-linear transformations of all submanifolds in the space of quadrics. In 1979 Morimoto [17] announced a number of statements on classification of MongeAmpere equations of a special form based on the theory of G-structures. Our approach to the classification problem is based on a relation between the differential forms on the manifold JM of 1-jets of smooth functions and Monge-Ampere equations [10]. We rely much upon [10]. Note also that the classification problem is in close connection with certain problems of the classical invariant theory which are, in our opinion, of an independent interest. One of them is the problem of description of orbits of the natural action of the symplectic group in the space of exterior forms. Notice that the analogy with differential equations enables us to understand better certain problems of the invariant theory for this group. In particular, this concerns the description of Sp-orbits of ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE. 0012-9593/93/03 281 28/$ 4.80/ © Gauthier-Villars 282 V. V. LYCHAGIN €t dl. 3-forms on the 6-dimensional real symplectic space and also certain orbits of 4-forms on the 8-dimensional space. The text consists of 6 sections (S). In Section 0 we introduce basic concepts and constructions used below. The main source of definitions and ideas is paper [14]. In S. 1 we consider the classical two-dimensional problem of S. Lie on local classification of Monge-Ampere equations. We start with the algebraic model of our problem and study a symplectic equivalence of 2-forms in 2 ̂ -dimensional space. The normal forms are listed. The non-degenerate two-dimensional Monge-Ampere equation determines an additional geometric structure on 4-dimensional symplectic manifold. The elliptic equations define an almost-complex structure and hyperbolic—an almost-product structure. The Newlander-Nirenberg theorem (in the elliptic case) and Frobenius theorem in a hyperbolic one give necessary and sufficient conditions of equivalence of our Monge-Ampere to a constant coefficients equation. At the end of the section we prove two classic S. Lie theorems on the reduction of non-linear Monge-Ampere to a quazilinear one and on the normal forms of MongeAmpere admitting an intermediate integral. Sections 2-5 are devoted to the algebraic problems arising in a high-dimensional (n ̂ 3) classification. In S.2 we solve the problem of symplectic classification of effective 3-forms on 6-dimensional real space. We list normal forms and indicate that even in the 3-dimensional case the Monge-Ampere equations corresponding to generic orbits are not linearizable even at a point. Theorem 2.6 generalizes the corresponding results ofJ.-I. Igusa [8] and V. Popov [19] in the case of an algebraically non-closed field. Moreover we directly built an invariant of the classification problem. In the next section (S. 3) we give a short outline of the description procedure for normal forms in the dimension greater than 3 (Theorem 3.4). In theorem 3.5 the normal forms of effective 4-forms on 8-dimensional symplectic space are listed (under some natural conditions). We establish a relation between the set of all transvections admitted by a given form and the symplectic classification of effective forms. After that the stabilizers of effective /2-forms are described. We also calculate the stationary subalgebras of the most important types of effective forms. In S. 4 we make an algebraic digression and study the finiteness conditions on type of effective forms stabilizers. First we classify reductive subalgebras / in EndV with nontrivial first Cartan prolongation. If the representation / -> EndV is irreducible the results are known (Theorem 4.2.1). Theorem 4.3.1 solves this problem for reducible representations. Then we study stabilizers. Theorem 4.4.1 states the general result on finiteness of the stationary subalgebra of a regular element. We also give several reformulations and corollaries of this theorem for stabilizers of effective forms under a 4° SERIE TOME 26 1993 N° 3 MONGE-AMPERE EQUATIONS 283 symplectic action. At the end of the section we explicitly calculate Cartan prolongations of the stabilizers of several important types of effective forms. In S. 5 we study involutiveness of the stabilizers of effective forms. The importance of these questions to the classification problem is explained in 6 where we identify the symbol of a Monge-Ampere equation corresponding to the homology equation of the classification problem with the stabilizer of the corresponding form. The involutiveness of the symbol is one of the conditions of the criterion for formal integrability. S. 6 is the central one from the classification problem viewpoint. Theorem 6.4.1 gives conditions for reducibility of an equation with analytic coefficients by an analytic symplectic diffeomorphism to an equation with constant coefficients in R\". The finiteness of stabilizer condition enables us to strengthen this theorem and generalize it to C°°-setting (Theorem 6.6.1). At the end of the section Theorem 6.6.1 is applied to the classification of effective forms and the corresponding Monge-Ampere equations on the 3-dimensional manifolds. The main results were published in [5, 15, 16]. 0. Formulation of the problem 0.1. Let (V, Q) be a symplectic space over R with the structure form OeA^V*) and dim V =2n. Denote by r:V-^V* the isomorphism determined by the structure form 0 i. e. V (X) = ;x (0) and by F,: A (V) -^ A (V*) its exterior powers, F, = A (F). For every coeA^V*) denote by z^eA^V) the svector corresponding to co, i.e. r,(z;J=©. In the algebra of exterior forms A*(V*)= © A^V*) introduce two operators s^o T^CV*)-^^^*), the operator of exterior multiplication by the structure form Q T(O))=CO A Q, and 1 ̂ (V*)-^\"^*), the operator of inner multiplication by the canonical bi vector v^, -L (o)) = ̂ (co). An exterior form coeA^V*), k^n will be called effective if -Lco=0 or, equivalently, coeA^V*) is effective if and only ifT^o)=0 for s=n-k, T^O/^T [14]. 0.2. Let M be a smooth manifold, J (M) the manifold of 1-jets of smooth functions on M, UeA^^M) the universal 1-form on 3M which determines the contact structure [12]. At each point x e J M the restriction of dV^ onto C (x) = Ker U^ determines a symplectic structure and therefore the operators ^A^C*^))-^^^*^)) and ^A^C*^))-^\"^*^)). The tangent space T^M) splits into the direct sum T^.^lV^CC^OIRXi where Xi is the contact vector field with the generating function 1 [12]. Therefore, if A^C*) the space of differential .y-forms on J M degenerated along Xi, (A^C*))^ is naturally identified with A^C*^)) and, besides, we have A (J M) = A (C*) © A(C*). We will say that (oeA^C*) is an effective form on J M if 10=0. Denote by A^ the set of all the effective ^-forms. ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE 284 V. V. LYCHAGIN €t al. 0.3. For every differential n-form (oeA^.PM) determine a non-linear differential operator ̂ acting via the formula A^ (h) =7\\ (h)* (co), A^: C (M) -> A\" (M) where j\\ (h) is the section determined by a function AeC°°(M). Two differential forms (D^, 0)3 € A\" (J M) determine the same operator if and only if co^ — o)^ € V^ where 1^ is the w-th homogeneous component of the ideal 1̂ <= A* (J M) formed by the elements of the form ®i A U + ©2 A d\\J. Since A^ (J M) ̂ A (J M)/I^ then Aco is determined by the effective part of the projection ofo) onto A^C*). In what follows we will assume that Ao is given by an effective form o). We will call the operators A^:C°°(M) ^A\"(M) the Monge-Ampere operators. 0.4. Determine the action of the group Ct(J M) of contact transformations of J M onto the Monge-Ampere operators setting a(AJA^) for aeCt(J M). Similarly define the action of the Lie algebra ct^M) of contact vector fields on J^M setting X^A^J^LX^CO); here Lx is the Lie derivative along X and Xy is a contact vector field with a generating function/eC (J M) [12]. 0.5. We will be interested in the problem of local classification of Monge-Ampere operators (equations) with respect to the group Ct(m) of the germs of contact diffeomorphisms preserving a point w. Hereafter we will assume that in a neighbourhood ofw there exists an infinitesimal contact symmetry Xp where f(m) 7^0, ofA^. Then there exists a local contact diffeomorphism sending Xy to X^ so that we may assume that L^ (co)=0. This means that co can be considered as a form on T*M and the classification problem for operators (equations) given by such forms as a classification problem of differential ^-forms on T* M wit