Group invariant solution for a free jet on a hemi-spherical shell

When two circular jets impinge upon each other along the axis of a hemi-spherical shell then a free jet on a hemi-spherical shell is formed. The governing equations are Prandtl’s momentum boundary layer equation and the continuity equation. The conserved quantity is required for the free jet on a hemi-spherical shell. The conserved quantity for the free jet on a hemi-spherical shell is established with the help of a conserved vector. The group invariant solution for the third-order partial differential equation for the stream function is constructed.     

Relative Equilibria of Molecules

Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0 . Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium configuration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each reflection plane there is a family of relative equilibria for which the molecule rotates about an axis perpendicular to the plane. We also show that if the equilibrium is nondegenerate and stable, then the minima, maxima, and saddle points of h correspond respectively to relative equilibria which are (orbitally) Liapounov stable, linearly stable, and linearly unstable. The stabilities of the bifurcating branches of relative equilibria are computed explicitly for XY ₂ , X ₃ , and XY ₄ molecules. These existence and stability results are corollaries of more general theorems on relative equilibria of G -invariant Hamiltonian systems that bifurcate from equilibria with finite isotropy subgroups as the momentum is varied. In the general case, the function h is defined on the Lie algebra dual {\frak g} ^* and the bifurcating relative equilibria correspond to critical points of the restrictions of h to the coadjoint orbits in {\frak g} ^* . Received June 9, 1997; second revision received December 15, 1997; final revision received January 19, 1998     

The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the x₃‐axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the x₃‐axis, we consider the reduced Hamiltonian associated with the total momentum along the x₃‐axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the x₃‐axis are sufficiently small. We determine the absolutely continuous spectrum of the reduced Hamiltonian and, when the ground state is simple, we prove that the renormalized mass of the dressed electron is greater than or equal to its bare one. We then deduce that the anomalous magnetic moment of the dressed electron is nonnegative. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamics of turbulent wake with small excess momentum in stratified media

A numerical model of turbulent wake dynamics with small excess momentum in linearly stratified fluid has been created on the basis of a three-dimensional parabolized system of averaged equations for motion, continuity, incompressibility, energy of turbulence and the rate of dissipation transfer. The calculation results show that as in the case of homogeneous fluid, the wake’s excess momentum of the order of ±5% and ±10% off the total momentum in the drag wake has a weak influence on the decay of the turbulent energy and more significant influence on the axis value of the defect of the longitudinal velocity component. Turbulent wakes with small excess momentum generate internal waves which differ only insignificantly from the waves generated by momentumless wakes.     

The steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability

The positions of relative equilibrium of a satellite carrying a two-degree-of-freedom powered gyroscope, in the axes of the framework of which only dissipative forces can act, are investigated within the limits of a restricted circular problem. For the case when the “satellite - gyroscope” system possesses the property of a gyrostat and the axis of the gyroscope frame is directed parallel to one of the principal central axes of inertia of the satellite, all the equilibrium positions are found as a function of the magnitude of the angular momentum of the rotor. It is established that the minimum number of equilibrium positions is equal to 32 and, in certain ranges of values of the system parameters, it can reach 80. All the positions satisfying the sufficient conditions for stability are also determined. The number of them is either equal to 4 or 8 depending on the values of the system parameters.     

Symmetric excited states for a mean-field model for a nucleon

In this paper, we consider a stationary model for a nucleon interacting with the ω and σ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit. By a shooting method, we prove the existence of infinitely many solutions with a given angular momentum. These solutions are ordered by the number of nodes of each component.     

The motion of an absolutely rigid body on a two-degrees-of-freedom joint in a uniform gravitational field

The motion of an absolutely rigid body attached to a fixed base by a two-degrees-of-freedom joint in a uniform gravitational field parallel to the fixed axis of the joint is studied qualitatively. Various kinds of motion are described and analysed, depending on the total mechanical energy and the projection of the angular momentum of the body onto the fixed axis of the joint as well as on the inertial parameters of the system.

This paper is a continuation of [1].     

The number of spanning trees of plane graphs with reflective symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line (called symmetry axis). Let G be a symmetric plane graph. We prove that if there is no edge in G intersected by its symmetry axis then the number of spanning trees of G can be expressed in terms of the product of the number of spanning trees of two smaller graphs, each of which has about half the number of vertices of G.     

The relativistic theory of chemical binding

Starting from Breit’s relativistic equation for a system of two electrons, it is shown that for a hydrogen molecule (or for a system of two electrons moving in a field of cylindrical symmetry) the component of the total angular momentum (J _x ) along the axis of the molecule (axis of symmetry) is a constant of motion. Thus every eigenstate of the system is simultaneously an eigenstate of J _x also, and a state of the system will specify, besides its energy, only the eigenvalue of the component of the angular momentum parallel to the axis of symmetry. The form of the four large components of the wave function relating to their dependence on the azimuthal co-ordinates has been given. The case of Russel-Saunders approximation has been considered in detail and the nature of the components of the wave function for the singlet and triplet states has been discussed. It is shown that the wave function for the ground state of the hydrogen molecule could be expressed as a sum of a set of symmetric functions of which the first term is the Heitler-London function, and that the wave function for a triplet state should be a superposition of anti-symmetric molecular orbitals. It is shown that relativistic theory brings about in a natural manner the facts relating to the ground state of the molecules C₂ and O₂. Finally, some remarks are made concerning the case of molecules for which the spinorbit and the spin-spin couplings are strong.     

Degasperis-Procesi方程解的渐进性质

主要研究Degasperis-Procesi(DP)方程强解的渐近性质,即通过对其强解的动量密度用渐近密度的方法,并在渐近密度唯一的假定下,证实了DP方程的正动量密度的渐进密度是支集在正轴上的Dirac测度的组合,且当时间趋于无穷时,动量密度集中在不同速度向右移动的小区域中.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope

For a dynamically symmetrical satellite carrying a two-degree-of-freedom powered gyroscope, all the relative equilibria in a circular orbit are found as a function of the angular momentum of the rotor and the angle between the precession axis of the gyroscope and the plane of dynamical symmetry. The case with no spring on the axis of the gyroscope frame and the case with a spring whose stiffness satisfies definite conditions are considered. The secular stability of the equilibria is investigated. For a system with dissipation in the axis of the gyroscope frame, the Barbashin–Krasovskii theorem is used to perform a detailed analysis, which enables the character of the Lyapunov stability of all the equilibria to be determined, with the exception of a few points. The results of a numerical solution of the problem of the optimal values of the system parameters, for which asymptotically stable equilibria are obtained with maximum speed, are presented.     

On rotating black holes in equilibrium in general relativity

We study asymptotically flat axially symmetric stationary solutions of the Einstein vacuum equations. These represent rotating black holes in equilibrium. The equations reduce outside the axis of symmetry to a harmonic map problem into the hyperbolic plane, with prescribed rates of blow-up for the map on the axis and at infinity as boundary conditions. We prove existence and uniqueness of solutions in the case of zero total angular momentum.     

Deformable Asymmetric Tops under Similarity Transformations

In this paper the configurations of relative equilibrium are determined for a deformable heavy top with a fixed point constrained to deform only by similarity transformations. We analyze the nonlinear orbital stability using the reduced energy–momentum method for configurations such that the center of the mass vector is parallel to the axis of gravity and coincides with an eigenvector of the Euler tensor. In addition, we study these conditions of stability for a specific deformable top given by a hyperelastic material of the Saint Venant–Kirchhoff type, and describe how these conditions are particularized in the special case of almost rigid tops.     

Three dimensional steady subsonic Euler flows in bounded nozzles

The existence and uniqueness of three dimensional steady subsonic Euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the entrance are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic–sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli function are both small, the existence and uniqueness of subsonic Euler flows with non-zero vorticity are established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div–curl system, and delicate estimate for the transport equations.     

The One-dimensional Fractional Supersymmetric Quantum Mechanical Operator of Momentum

In the case of the quantum generalization of stable Lévy processes, expressions for the Hermitian operator of momentum and its eigenfunctions are proposed. The normalization constant has been determined and its relation to the translation operator is shown. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer dimensionality α. The simplest nonlocal superalgebra is introduced.      

Balance laws for liquid crystal mixtures

Equations of balance of mass, linear momentum, angular momentum and energy are obtained for each of the species in a chemically reacting mixture of liquid crystals. The forms of these equations for the mixture as a whole are then arrived at.     

Dynamic modeling of a Stewart platform using the generalized momentum approach

Dynamic modeling of parallel manipulators presents an inherent complexity, mainly due to system closed-loop structure and kinematic constraints.In this paper, an approach based on the manipulator generalized momentum is explored and applied to the dynamic modeling of a Stewart platform. The generalized momentum is used to compute the kinetic component of the generalized force acting on each manipulator rigid body. Analytic expressions for the rigid bodies inertia and Coriolis and centripetal terms matrices are obtained, which can be added, as they are expressed in the same frame. Gravitational part of the generalized force is obtained using the manipulator potential energy. The computational load of the dynamic model is evaluated, measured by the number of arithmetic operations involved in the computation of the inertia and Coriolis and centripetal terms matrices. It is shown the model obtained using the proposed approach presents a low computational load. This could be an important advantage if fast simulation or model-based real-time control are envisaged.     

Countably Periodic Gibbs Measures of the Ising Model on the Cayley Tree

We diagonalize the metric Hamiltonian and evaluate the energy spectrum of the corresponding quasiparticles for a scalar field coupled to a curvature in the case of an N-dimensional homogeneous isotropic space. The energy spectrum for the quasiparticles corresponding to the diagonal form of the canonical Hamiltonian is also evaluated. We construct a modified energy–momentum tensor with the following properties: for the conformal scalar field, it coincides with the metric energy–momentum tensor; the energies of the particles corresponding to its diagonal form are equal to the oscillator frequency; and the number of such particles created in a nonstationary metric is finite. We show that the Hamiltonian defined by the modified energy–momentum tensor can be obtained as the canonical Hamiltonian under a certain choice of variables.     

The problem of wave momentum,radiation pressure and other quantities in the case of plane motions of an ideal gas

Small vibrations and waves of an ideal fluid gas (a liquid or gas) are considered in the quadratic approximation. Actual values of the momentum density and the pressure and also their averaged values are obtained in a number of specific characteristic problems. It is shown that the momentum of an isolated wave with a mean density equal to the density of the unperturbed medium, and the radiation pressure are due to non-linearity of the system of equations, and that this wave has no momentum if its profile remains unchanged during its motion. The latter assertion is also true for finite-amplitude waves.