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The classical and generalized Euler numbers, reduced with respect to an odd modulus, are represented as sums of exponentials. From this representation there follow congruences modulo powers of an odd prime p between elements of the Euler-Bernoulli triangles and the values of certain polynomials in two variables on sublattices with step p–1.

In the sequel without further notice all objects, such as manifolds, functions, mappings, distributions, vector fields, forms, etc., are supposed to be infinitely differentiable.

“The main thing we must do with the differential equations of physical models is to investigate what is possible and what it is necessary to change in them”, said Poincare (La valeur de la science, Part 2, Chapter V (Analysis and physics), p. 222 (1905b)). In 1931, following this prescription, A.A. Andronov (1933) came out with a vast program (also set forth in the preface to Andronov and Khajkin, 1937) which differs from the contemporary program of catastrophe theorists only in that the qualitative theory of differential equations and Poincare’s theory of bifurcations take the place of Whitney’s theory of singularities of differentiable mappings, as yet uncreated at his time.

Poincare already used “putting into general position” almost as is done today. For example, his classification of the singular points of generic vector fields on the plane (saddle, node, focus) is the prototype for many classifications in catastrophe theory. The theory of bifurcations of periodic solutions, also created by Poincare, anticipated the general theory of bifurcations, not only with respect to results, but also (and particularly) with respect to methods. Already in his dissertation (1879) (solving a problem of classification of differential equations posed by Darboux, who was the discover of the normal form of a symplectic structure) created the general method of normal forms, which leads to the classification of catastrophes if one applies it to functions instead of to differential equations.

It is well known that the projective space ℂP n carries a canonical symplectic structure (see, for instance, Arnol’d [1974] or Arnol’d and Givental’ [1985]). We consider a representation ρ: G →GL(V) of a Lie group G on a finite-dimensional complex vector space V. There is an induced action of G on the projective space P(V) associated with V. We denote by G(z) the orbit of this action passing through a point z ϵ P(V). It is now natural to ask when the symplectic structure of P(V) induces a symplectic structure on the orbits of G in P(V). Let us assume that G is compact, let T be a maximal torus in G, let g and t be the Lie algebras of G and T, respectively, and let L ℂ denote the complexification of a vector space L. We then have the root space decomposition \( {g^C} = {t^C} + \sum\limits_{\alpha \ne 0} {\mathbb{C}{E_\alpha }{H_\alpha }} = \left[ {{E_\alpha }{E_{ - \alpha }}} \right] \) (see, for instance, Helgason [1962], Jacobson [1962] or the Appendix). Let n: V\0 → P(V) be the natural projection which maps a vector υ ∊ V\0 into the straight line passing through υ and the origin.

Bifurcations of phase portraits in the neighborhood of a cycle are completely described by bifurcations of the corresponding monodromy transformations. Therefore the basic objects of study in this chapter are the bifurcations of germs of diffeomorphisms at a fixed point. Local families of germs of diffeomorphisms, their equivalence and weak equivalence, and induced and versal deformations of such germs are defined just as for germs of vector fields (see Sect. 1.5 of Chap. 1). Analogs of the Reduction Theorem are also true for germs of diffeomorphisms at a fixed point; see Sect. 1.6 of Chapter 1 and Arnol’d and Il’yashenko (1985, Sect. 2.4 of Chap.6). The restriction of a germ of a diffeomorphism to its center manifold is called the reduced germ of the diffeomorphism. We note that a reduced germ can change orientation, even if the original germ did not: for example, \( x \mapsto {\text{diag(1;2; - }}\frac{1}{2}{\text{)}}x,{\text{ }}x \in \mathbb{R}^4 . \). The bifurcations of germs of diffeomorphisms are described below, and then the results obtained are translated into the language of differential equations.

In this chapter we consider the simplest nonholonomic variational problems. We study three-dimensional nonholonomic Lie groups, i.e. groups with a left-invariant nonholonomic distribution. Our main subject is the study of the nonholonomic geodesic flow (NG-flow), more precisely, of the nonholonomic sphere, of the wave front (Section 1), and of the general dynamical properties of the flow (Section 2). The mixed bundle for Lie groups is the direct product G × (V ⊕ V⊥). In Section 1.1 we show that the NG-flow on the mixed bundle is the semidirect product with base V ⊕ V⊥ and fiber G. In Section 1.2 we describe left-invariant metric tensors on Lie algebras; in Section 1.3 the normal forms for the equations of nonholonomic geodesics are obtained. In Section 1.4 we study the reduced flow on V ⊕ V ⊥. In the subsequent Sections (1.5–1.7) we describe local properties of the flow on the fiber; in Section 1.5 we describe the e-wave front of the NG-flow and the e-sphere of the nonholonomic metrics which appear to be manifolds with singularities, the same for all three-dimensional nonholonomic Lie groups. In Section 1.6 we describe their topology and in Section 1.7 their metric structure.

At first glance, the most natural classification principle is classifying by codimension beginning with small codimensions. To classify objects “up to codimension ⩽ k” means to represent the entire space of objects studied as a finite union of submanifolds of codimensions not greater than k (called classes) and a remainder of codimension ⩾k + 1 so that within each class an object’s properties that are of interest to us do not change. Then all objects in typical, no more than k-parameter families, belong to our classes: the remaining ones may be avoided by a small perturbation of the family.

In 1654 Huygens (1673) constructed the theory of evolutes and involutes of plane curves. He noted that an involute or an evolute of a smooth curve has cusps of semi-cubical type. The evolute of a smooth curve is a caustic (the envelope of the family of rays orthogonal to the curve); see Fig. 6. (Leonardo da Vinci (see Bennequin (1986)) had already investigated caustics; the term was apparently introduced by Tschirnhaus (1682)). The family of involutes of the caustics in Fig. 6 is shown in Fig. 7. It is the family of wave fronts corresponding to the same family of rays. (All of this was also discovered by Huygens.) Thus, Huygens essentially discovered the stability of cusps on caustics and wave fronts. Nowadays these singularities are connected with the pleats of the corresponding smooth mappings, and they belong to the most important applications of catostrope theory. Fig. 7 was discussed by Cayley (1868b).

Physicists have always made use of constructions more or less equivalent to catastrophe theory in investigations of concrete problems. In this sense, catastrophe theory may be compared with mathematical analysis. Without the help of analysis, Huygens was able to solve the majority of problems solved by Newton. But such solutions required the genius of Huygens while, nowadays, the same problems may be solved with the help of analysis by any student. In exactly the same way, mastery of the technique of singularity theory allows one to obtain results automatically that otherwise require inventiveness and substantial efforts of imagination, simultaneously extending them to more complicated situations where “elementary” methods would lead to vast calculations. (1974) in an article “Nobel prizes for catastrophes” points out many examples of acknowledged physical achievements, whose authors, independently of one another, used ideas formalized later in singularity theory.

In the middle of the 1960s R. Thom, the first person to appreciate the value of H. Whitney’s theory of singularities for applications, undertook to apply it to problems of biology (namely, to embryology), and broadly to problems of morphogenesis in general.

A pair (M, ω) consisting of a 2n-dimensional manifold M together with a closed 2-form ω is called a symplectic manifold if the form ω is nondegenerate, i.e. if ω n = ω ∧ · ... · ω ≢ 0.

In this section we consider the nonholonomic variational problem on the minimum of length on a Riemannian manifold with constraints given by a nonholonomic distribution. The solutions to this problem, the nonholonomic geodesics, satisfy the Euler-Lagrange equations of a conditional problem. They generate a nonholonomic geodesic flow defined on the mixed bundle which is the direct sum of the distribution and its annihilator in the cotangent bundle (see Section 1.3). This flow allows to extend to the nonholonomic case the notion of exponential mapping.

The name of Newton and his enormous contribution to both mathematics and physics are well known. He was born in 1642, in the year of Galileo’s death, and died in 1727. The work of Newton in the field of the theory of gravitation became famous in continental Europe thanks to Voltaire, who paid a visit to England in the last years of Newton’s life and propagated the law of universal gravitation, which made a great impression on him. Voltaire informed the world about the famous apple, which Newton’s niece Catherine Barton had told him about (2).

In the Principia there are two purely mathematical pages containing an astonishingly modern topological proof of a remarkable theorem on the transcendence of Abelian integrals (42).

The hierarchy of formsf(dx 1∧...∧dx n )α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogeneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

Evolution of initially smooth perturbations in a cold self-gravitating medium in a Friedmann Universe gives rise to the formation of singularities in the distribution of density in a manner similar...

A coordinate-free exposition of the theory of multidimensional Jacobi coordinates, which are used to integrate the equations of geodesics on an ellipsoid as well as other equations, is given. Generalizations of the theorems of Newton and Ivory, concerning the attraction field produced by an ellipsoid, are given.

CONTENTSIntroduction ??1. Rayleigh-Schr?dinger series for perturbations of a simple eigenvalue ??2. Perturbations of a multiple eigenvalue ??3. Widths of forbidden zones of even Hill's equations of Mathieu type ??4. Formulae for the denominators of terms in the perturbation theory series ??5. Conclusion of the investigation of forbidden zones ??6. Widths of resonance zones for mappings of the circleReferences

The authors describe the critical values of the maps at time''t'' and their evolution as ''t'' changes for potential initial velocity fields in general position under the assumption that the force field is potential. The paper is concerned with the structure and evolution of caustics of a general one-parameter family of Lagrangian maps of manifolds of dimension not exceeding three. For each type of evolution, the authors give a detailed geometric description of the structure of the singularity. The investigation required new algebraic information about the manifold of polynomials with multiple roots; these are given in the paper.

CONTENTSIntroduction § 1. Is there such a thing as symplectic topology? § 2. Generalizations of the geometric theorem of Poincare § 3. Hyperbolic Morse theory § 4. Intersections of Lagrangian manifolds § 5. Legendrian submanifolds of contact manifolds § 6. Lagrangian and Legendrian knots § 7. Two theorems of Givental' on Lagrangian embeddings § 8. Odd-dimensional analogues § 9. Optical Lagrangian manifoldsReferences