Kirchhoff Problems Research Articles

A new family of phase patterns in three-dimensional dynamics of a solid body interacting with matter

1. Setting the problem. At the present time, solving the problem of motion of a solid body in a resisting matter completely depends on developing mathematical methods and possible model constraints inherent in the problem. For example, the well-known Kirchhoff problem of motion of a solid body in a perfect incompressible fluid, being quiescent at infinity and executing vortex-free motion [1], deals with only one aspect of the problem. This aspect is related to the problem of integrability of corresponding dynamic systems (in our case, with the existence of the complete set of analytic and meromorphic first integrals). It is easy to note that the Kirchhoff problem is one of the first approximations to the description of the interaction between a body and a medium because, on introducing an arbitrarily small viscosity, the dynamic Kirchhoff systems cease to be conservative. Thus, asymptotic limiting sets appear in the phase space of resulting systems because they become dissipative “as a whole.” Therefore, no complete set of even-continuous first integrals can be asserted [2]. In connection with this comment, we indicate another aspect of the problem, namely, the complete qualitative analysis of dynamic systems (the topology of phase-space splitting into trajectories). It is the aspect that is an object of the present analysis. 2. Initial conditions and choice of variables. We consider a problem of a three-dimensional motion of a dynamically symmetric solid body under the condition that the line of action of the force S applied to the body from the matter does not change its orientation with respect to the body, a part of the body surface having the shape of a flat disk. The matter flows around the disk [3] according to laws of a jet flow around a body [4, 5]. The force S is aligned with the normal to the disk and is a quadratic form of the velocity of the disk center. The gravity force acting on the body is assumed to be negligible compared to the resisting force of the matter. The choice of six dynamic phase variables v , α , β (i.e., spherical coordinates of the velocity vector for the disk center) and p , q , r (the components of the body absolute angular velocity in the coordinate system linked with the body) allows us to consider the sixorder system of dynamic equations as independent. Moreover, because the resisting force admits a group of the body rotation about the axis of dynamic symmetry (which passes through the center of mass and the center of the disk), the longitudinal component of the angular velocity being conserved: p = p 0 = const [3, 6]. 3. Dynamic equations of motion. If ( A , B , B ) are the principal moments of inertia for the body, m and σ are its mass and the distance between the center of mass and the disk, respectively, z 1 = q cos β + r sin β , z 2 = r cos β — q sin β , z i = Z i v , i = 1, 2, = α ’ v , = β ’ v ,

Factor spaces and implications on Kirchhoff equations with clamped boundary conditions

We consider mixed problems for the Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped boundary conditions BC (clamped control). If w denotes the elastic displacement and θ the temperature, we establish sharp regularity of {w, wt, wtt} in the elastic case, and of {w, wt, wtt, θ} in the thermoelastic case. Our results complement those by Lagnese and Lions (1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary homogeneous cases. The passage from the boundary homogeneous cases to the corresponding mixed problems involves a duality argument. However, in the present case of clamped BC, and only in this case, the duality argument in question is both delicate and technical. In this respect, the clamped BC are “exceptional” within the set of canonical BC (hinged, clamped, free BC). Indeed, it produces new phenomena which are accounted for by introducing new, untraditional factor (quotient) spaces. These are critical in describing both interior regularity and exact controllability of mixed elastic and thermoelastic Kirchhoff problems with clamped controls.

Doubly asymptotic trajectories of Lagrangian systems in homogeneous force fields

We study Lagrangian systems with symmetry under the action of a constant generalized force in the direction of the symmetry field. After Routh's reduction, such systems become nonautonomous with Lagrangian quadratic in time. We prove the existence of solutions tending to an orbit of the symmetry group as t→± ∞. As an example, we study doubly asymptotic solutions for the Kirchhoff problem of a heavy rigid body in an infinite volume of incompressible ideal fluid performing a potential motion.

On attainability of Hashin-Shtrikman bounds by iterative hexagonal layering. Plane elasticity problems

THE PAPER presents a derivation of the Hashin-Shtrikman bounds for the plane elasticity problems and for the Kirchhoff problem of plates in bending. A two-dimensional counterpart of the method of FRANCFORT-MURAT [1] is applied. The method consists of three subsequent layerings along the directions of vertices of a unilateral triangle.

Including near-field terms in the isochrone integration method for application to finite-fault or kirchhoff boundary integral problems

In many types of seismological and engineering problems, it is necessary to perform a surface integral involving a Green's function having either a source or receiver point on the surface. For far-field terms of the Green's function, it is well known that the surface integral can be reduced to a line integral, yielding a considerable computational advantage. In this work we show that a similar transformation to a line integral can be made for the near-field terms when the Green's function is for a uniform whole space. The necessary condition is that the various terms in the Green's function can be transformed into sums of nondispersive pulses. We accomplish this for near-field terms in a whole space by repeatedly time-differentiating the terms.

Bifurcation of stationary vortex configurations: II. Topology and integrability

We study the dynamics of finitely many vortices in a circular disk and compare the integrability of this problem with that of Kirchhoff's problem of vortices in the plane. The effect of the topology of the phase space on the two Hamiltonian systems is compared. Our goal is to apply topological methods uniformly to investigate the flow of these dynamical systems.

Calculation of the transient acoustic field at the surface of a rigid body

The integral equations which describe the transient acoustic field at the surface of a rigid body immersed in an infinite inviscid fluid are derived and reduced to an equivalent matrix representation by the method of collocation. The method is then applied to Kirchhoff's problem of an impulsively accelerated rigid sphere.