New Families of Conservative Systems on S2 Possessing an Integral of Fourth Degree in Momenta

There is a well-known example of an integrable conservative system on S2, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. In this paper we propose new families of examples of conservative systems on S2 possessing an integral of fourth degree in momenta.     

The topology of the analog of Kovalevskaya integrability case on the Lie algebra <Emphasis Type="Italic">so</Emphasis>(4) under zero area integral

The topology of the space of closures of solutions to an integrable system on the Lie algebra so(4) being an analogue of the Kovalevskaya case has been studied. Fomenko-Zieschang invariants are calculated for this purpose in the case of zero area integral, which classify isoenergetic 3-surfaces and the corresponding Liouville foliations.     

ON SOLUTION OF GYROSCOPE SYSTEM IN GENERAL KOVALEVSKYAYA CASE

The solutions of gyroscope system in general Kovalevskava case(A=B= 2C,z_G=0)is studied.Based on the four independent first integrals of gy- roscope system in general Kovalevskava case,it is proved that the solutions of gyroscope system in this case can be expressed by hyper-elliptic functions.     

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

We consider the Euler equations on the Lie algebra so(4, ℂ) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. This is a consequence of more general fact that for these systems the existence of Darboux polynomial with no vanishing cofactor implies the existence of polynomial fourth integral.     

Bifurcations of completely integrable 2-variable first-order partial differential equations

In this paper, we consider an implicit 2-variable first-order partial differential equation with complete integral. As an application of the Legendrian singularity theory, we give a generic classification of bifurcations of such differential equations with respect to the equivalence relation which is given by the group of point transformations following S. Lie?s view. Since two one-parameter unfoldings of such differential equations are equivalent if and only if induced one-parameter unfoldings of integral diagrams are equivalent for generic equations, our normal forms are represented by one-parameter integral diagrams.     

Classical perturbation theory and resonances in some rigid body systems

We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincaré sets are similar to the ones of the Kovalevskaya case.     

When nonautonomous equations are equivalent to autonomopus ones

We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit constant solutions to which there correspond non-constant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.     

用两个单参数Lie群求三阶自治系统的积分因子

给出并证明了利用三阶自治系统接受的两个单参数Lie群的生成元求系统积分因子的力法.     

Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.     

On the embeddability of certain infinitely divisible probability measures on Lie groups

We describe certain sufficient conditions for an infinitely divisible probability measure on a Lie group to be embeddable in a continuous one-parameter semigroup of probability measures. A major class of Lie groups involved in the analysis consists of central extensions of almost algebraic groups by compactly generated abelian groups without vector part. This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain connected Lie groups, including the so called Walnut group. The embeddability is concluded also under certain other conditions. Our methods are based on a detailed study of actions of certain nilpotent groups on special spaces of probability measures and on Fourier analysis along the fibering of the extension.     

一个Boussinesq系统的单参数不变群与孤子解

研究一个Boussinesq系统的不变群的向量场及其构成的李代数,并利用单参数不变群,由该系统的解去生成一些新的孤子解.     

利用Lie群方法求Burgers-Huxley方程行波类首次积分

本文运用Lie群理论,证明了Burgers-Huxley方程的行波解所满足的二阶非线性方程在参数满足一定关系时,在经典意义下接受一个两参数Lie群,此时可用积分法求其首次积分.     

Bifurcation of common levels of first integrals of the kovalevskaya problem

The structure of integral manifolds in the Kovalevskaya problem of a heavy solid about a fixed point is considered. An analytic definition of a bifurcation set is obtained, and bifurcation diagrams are constructed. The number of two-dimensional toruses that appear in the composition of the integral manifold is indicated for each connected component, additional to the bifurcation set in the space of first integral constants.     

Schatten classes on compact manifolds: Kernel conditions

In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.     

Groups of diffeomorphisms and geometric wraps of manifolds over ultra-normed fields

The article is devoted to the study of groups of diffeomorphisms and wraps of manifolds over ultra-metric fields of zero and positive characteristic. Different types of topologies are considered on groups of wraps and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies, pairwise incomparable ones are also found. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. Theorems on projective limit decompositions of these groups and their compactifications for compact manifolds are proved. Moreover, the existence of one-parameter local subgroups of diffeomorphism groups is proved.     

Embedding theorems for two-parameter groups of formal power series and related problems

The classical problem of analytic iteration is that of embedding analytic functions in one-parameter Lie groups of formal power series. The main purpose of the present paper is to consider similar problems for two-parameter groups. These problems are closely related to problems concerning conjugate power series and conjugate one parameter groups of such series, to which the first sections of the paper are devoted. As an application of the conjugacy theorems and the embedding theorems we bring an algebraic characterization of the class of the two-parameter groups.     

On a class of integrable systems with a quartic first integral

We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some detail and leads to a class of models on the manifolds {ie394-1}², ?² or ?². As special cases we recover Kovalevskaya’s integrable system and a generalization of it due to Goryachev.     

Refinement of Ado's Theorem in Low Dimensions and Application in Affine Geometry

In this article, we construct a faithful representation with the lowest dimension for every complex Lie algebra in dimension ≤ 4. In particular, in our construction, in the case that the faithful representation has the same dimension of the Lie algebra, it can induce an étale affine representation with base zero which has a natural and simple form and gives a compatible left-symmetric algebra on the Lie algebra. Such affine representations do not contain any nontrivial one-parameter subgroups of translation.     

Interaction of the folded Whitney umbrella and swallowtail in slow-fast systems

The graph of a first integral of a smooth slow-fast system with two slow variables is a singular surface in the three-dimensional space; the variation of an external parameter on which the system depends gives rise to perestroikas (=transitions) of this surface. We find a normal form and present figures of the perestroika that describes the interaction between the swallowtail and folded Whitney umbrella on the graph of a first integral of a generic one-parameter family of such systems.     

On reductive group actions and fixed points

Among analytic actions of reductive groups on projective varieties, we characterize the algebraic ones by the existence of fixed points for one-parameter subgroups. This applies to the problem of lifting the action of a compact Lie group on a projective manifold to a line bundle.