Small Motions and Normal Oscillations of a Double Pendulum with Cavities Containing a Viscous Incompressible Liquid

The linear hydrodynamic problem involving the small motions and normal oscillations of a double pendulum with cavities completely filled with a liquid is examined. The problem is solved using the methods of functional analysis. An existence theorem is formulated for the solutions to the Cauchy problem and the properties of the normal oscillations are described.     

Oscillations of a Stratified Liquid Partially Covered with Ice (General Case)

We study the problem of small motions of an ideal stratified liquid whose free surface consists of three regions: liquid surface without ice, a region of elastic ice, and a region of crumbled ice. The elastic ice is modeled by an elastic plate. The crumbled ice is understood as weighty particles of some matter floating on the free surface. Using the method of orthogonal projection of boundary conditions on a moving surface and the introduction of auxiliary problems, we reduce the original initial boundary value problem to an equivalent Cauchy problem for a second-order differential equation in a Hilbert space. We obtain conditions under which there exists a strong (with respect to time) solution of the initial boundary value problem describing the evolution of the hydrodynamic system under consideration.

     

A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

This article presents a rigorous existence theory for small-amplitude threedimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the timelike variable. Wave motions that are periodic in a second, different horizontal direction are detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom.Dedicated to Klaus Kirchgässner on the occasion of his seventieth birthday     

The Lyapunov-Poincaré method in the theory of periodic motions

The problem of the periodic motions of a system with a small parameter is solved. The non-rough cases, when the problem cannot be solved by a generating system obtained for a zero value of the small parameter, are investigated. Lyapunov's idea of using a new generating system which already contains the small parameter is systematically developed. Systems of general form, inverse systems and systems close to inverse are investigated.     

Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate

The motions of a non-autonomous Hamiltonian system with one degree of freedom which is periodic in time and where the Hamiltonian contains a small parameter is considered. The origin of coordinates of the phase space is the equilibrium position of the unperturbed or complete system, which is stable in the linear approximation. It is assumed that there is degeneracy in the unperturbed Hamiltonian when account is taken of terms no higher than the fourth degree (the frequency of the small linear oscillations depends on the amplitude) and, in this case, one of the resonances of up to the fourth order inclusive is realized in the system. Model Hamiltonians are constructed for each case of resonance and a qualitative investigation of the motions of the model system is carried out. Using Poincaré's theory of periodic motions and KAM-theory, a rigorous solution is given of the problem of the existence, bifurcations and stability of the periodic motions of the initial system, which are analytic with respect to fractional powers of the small parameter. The resonant periodic motions (in the case of the degeneracy being considered) of a spherical pendulum with an oscillating suspension point are investigated as an application.     

Non-linear oscillations of a Hamiltonian system in the case of 3:1 resonance

The motion of an autonomous Hamiltonian system with two degrees of freedom near its equilibrium position is considered. It is assumed that, in a certain region of the equilibrium position, the Hamiltonian is an analytic and sign-definite function, while the frequencies of linear oscillations satisfy a 3:1 ratio. A detailed analysis of the truncated system, corresponding to the normalized Hamiltonian is given, in which terms of higher than the fourth order are dropped. It is shown that the truncated system can be integrated in terms of Jacobi elliptic functions, and its solutions describe either periodic motions or motions that are asymptotic to periodic motions, or conventionally periodic motions. It is established, using the KAM-theory methods, that the majority of conventionally periodic motions are also preserved in the complete system. Moreover, in a fairly small neighbourhood of the equilibrium position, the trajectories of the complete system, which are not conventionally periodic, form a set of exponentially small measure. The results of the investigation are used in the problem of the motion of a dynamically symmetrical satellite in the region of its cylindrical precession.     

Stellar Hydrodynamic Equations for a Thin Disk Galaxy

Assuming that the motions of stars depart only slightly from circular, finite closed systems of stellar hydrodynamic equations are obtained for a disk galaxy by taking moments of the collisionless Boltzmann equation. The usual closure problem is avoided in that explicit expressions are obtained for the highest moments arising.     

Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.     

Problem of small and normal oscillations of a rotating elastic body filled with an ideal barotropic liquid

An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.     

一维流体动力学半导体方程解的Lp-收敛率

文讨论了一维流体动力学半导体方程,当压强函数为p(n)=kn~r,k >0,r≥1时,我们得到了带小初值的Cauchy问题的解的存在性.利用格林函数的办法,我们还得到解的L~p-估计,即当初值是某一常状态附近的小扰动时,其相应的解也是该常状态附近的小扰动。     

多个同频摇荡剖面引起的水面波辐射

在内场中使用简单Green函数的边界元方法与外场的速度势特征函数展开式相结合,用于求解多个同频摇荡剖面引起的水面波辐射问题的频域解·　方法适用于外场为定深的水域以及内场的复杂边界条件,各剖面的摇荡模态、幅值和相位可以互不相同·　利用摄动展开完整地求解了流场的二阶速度势和各个剖面所受的一、二阶水动力·　与单个剖面的情况相比,数值结果证实了多个剖面辐射引起的诸如水波共振和负附加质量等水动力干扰现象,这对于多体结构的锚泊系统和其它海洋工程设施的设计是很重要的·     

Oscillations of stratified fluids

We study the problem on small motions and normal oscillations of a system of two heavy immiscible stratified fluids partially filling a fixed vessel. The lower fluid is assumed to be viscous, while the upper one is assumed to be ideal. We find sufficient existence conditions for a strong (with respect to the time variable) solution of the initial-boundary value problem describing the evolution of the specified hydraulic system. For the corresponding spectral system, we obtain results about the localization of the spectrum, asymptotic behavior of branches of eigenvalues, and existence of the substantial spectrum of the problem.     

On Optimum Propulsion by Means of Small Periodic Motions of a Rigid Profile II. Optimization of the Period and Numerical Results

We present some numerically calculated optimum thrust generating small amplitude periodic motions of a rigid profile in an inviscid imcompressible fluid. The motions considered consist in general of both a heaving and a pitching part and have common period. Apart from the prescribed thrust, the motions have to satisfy the demand that the contribution to the total thrust of the suction at the leading edge not exceed a given number and are furthermore subjected to a constraint on their amplitude. Solutions of an analogous optimization problem for pure heaving motions are also discussed. Furthermore, the problem of optimizing the period of the motions is considered.     

The orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom in the case of 3:1 resonance

The problem of the orbital stability of periodic motions, produced from an equilibrium position of an autonomous Hamiltonian system with two degrees of freedom is considered. The Hamiltonian function is assumed to be analytic and alternating in a certain neighbourhood of the equilibrium position, the eigenvalues of the matrix of the linearized system are pure imaginary, and the frequencies of the linear oscillations satisfy a 3:1 ratio. The problem of the orbital stability of periodic motions is solved in a rigorous non-linear formulation. It is shown that short-period motions are orbitally stable with the sole exception of the case corresponding to bifurcation of short-period and long-period motions. In this particular case there is an unstable short-period orbit. It is established that, if the equilibrium position is stable, then, depending on the values of the system parameters, there is only one family of orbitally stable long-period motions, or two families of orbitally stable and one family of unstable long-period motions. If the equilibrium position is unstable, there is only one family of unstable long-period motions or one family of orbitally stable and two families of unstable long-period motions. Special cases, corresponding to bifurcation of long-period motions or degeneration in the problem of stability, when an additional analysis is necessary, may be exceptions. The problem of the orbital stability of the periodic motions of a dynamically symmetrical satellite close to its steady rotation is considered as an application.     

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.     

The periodic motions of a non-autonomous Hamiltonian system with two degrees of freedom at parametric resonance of the fundamental type

The motion of an almost autonomous Hamiltonian system with two degrees of freedom, 2π-periodic in time, is considered. It is assumed that the origin is an equilibrium position of the system, the linearized unperturbed system is stable, and its characteristic exponents ±iωj (j = 1,2) are pure imaginary. In addition, it is assumed that the number 2ω₁ is approximately an integer, that is, the system exhibits parametric resonance of the fundamental type. Using Poincaré's theory of periodic motion and KAM-theory, it is shown that 4π-periodic motions of the system exist in a fairly small neighbourhood of the origin, and their bifurcation and stability are investigated. As applications, periodic motions are constructed in cases of parametric resonance of the fundamental type in the following problems: the plane elliptical restricted three-body problem near triangular libration points, and the problem of the motion of a dynamically symmetrical artificial satellite near its cylindrical precession in an elliptical orbit of small eccentricity.     

Oscillations of Stratified Liquid Partially Covered by Crumpling Ice

We study the problem on small motions of ideal stratified fluid with a free surface, partially covered by crumbling ice. By the method of orthogonal projecting the boundary conditions on the moving surface and, with the help of investigation of some auxiliary problems, the original initial-boundary value problem is reduced to the equivalent Cauchy problem for a second order differential equation in a Hilbert space. We find sufficient existence conditions for existence of a strong (with respect to the time variable) solution to the initial-boundary value problem describing evolution of the specified hydrodynamics system.     

The global classical solution to compressible system with fractional viscous term

We consider the initial value problem to the fractional system of motions for compressible viscous fluids in this paper. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the large-time behavior of the solution, where solutions converge to a constant steady state exponentially in time.     

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.