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1.
The direct problem of dynamics in two dimensions is modeled by a nonlinear second-order partial differential equation, which is therefore difficult to be solved. The task may be made easier by adding some constraints on the unknown function = f y /f x , where f(x, y) = c is the monoparametric family of orbits traced in the xy Cartesian plane by a material point of unit mass, under the action of a given potential V(x, y). If the function is supposed to verify a linear first-order partial differential equation, for potentials V satisfying a differential condition, can be found as a common solution of certain polynomial equations.The various situations which can appear are discussed and are then illustrated by some examples, for which the energy on the members of the family, as well as the region where the motion takes place, are determined. One example is dedicated to a Hénon—Heiles type potential, while another one gives rise to families of isothermal curves (a special case of orthogonal families). The connection between the inverse/direct problem of dynamics and the possibility of detecting integrability of a given potential is briefly discussed.This revised version was published online in October 2005 with corrections to the Cover Date.  相似文献   

2.
This paper derives asymptotic expansions of ellipsoidal coordinates in Cartesian coordinates and an expansion in spherical harmonics of the dominant term for the solution of Laplace's equation corresponding to the gravitational force function for a two-dimensional finite body.On comparing the expansion of the dominant term derived here with known expansions of the force functions of the Earth's and Moon's gravitation the author obtains values for the semimajor axes and eccentricities of the singular ellipses of these bodies in terms of the second degree harmonic coefficientsc 20 andc 22.  相似文献   

3.
In this paper a slightly different approach is proposed for the process of determining the functions S m and H m * of the algorithm of the canonical version of Hori method. This process will be referred to as integration theory of the mth order equation of the method. It will be shown that the ordinary differential equation with an auxiliary parameter t * as independent variable, introduced through Hori auxiliary system, can be replaced by a partial differential equation in the time t. In this way, the mth order equation of the algorithm assumes a form very similar to the one of other perturbation methods. In virtue of this new approach of the integration theory for Hori method, Lagrange's variational equations introduced by Sessin are revised. As an example, the Duffing equation is solved through this new approach.  相似文献   

4.
Abstract— The thermodynamic properties of SiO2 are approximated over a range of pressures and temperatures important under the extreme conditions achieved in impacts at typical solar system velocities from 5 to about 70 km/s. The liquid/vapor phase curve and critical point of SiO2 are computed using the equation of state (EOS) program ANEOS. To achieve this goal, two shortcomings of ANEOS are corrected. ANEOS, originally developed at Sandia National Laboratories to describe metals, treats the vapor phase as a monatomic mixture of atoms, rather than molecular clusters. It also assumes a Morse potential for the expanded solid state. Neither of these assumptions is accurate for geologic materials, such as SiO2, that contain molecular clusters in the vapor phase and are better described by a Mie‐type potential in the solid state. Using the updates described here, an EOS adequate for numerical hydrocode computations is constructed that agrees well with shock data at pressures up to at least 600 GPa and temperatures up to 50,000 K. This EOS also gives a good representation of the liquid/vapor transition at much lower pressures and temperatures. The estimated critical point parameters for SiO2 are Pc = 0.19 GPa, Tc = 5400K, ρc = 550 kg/m3.  相似文献   

5.
In this work, which is a supplemental to previous one, we undertake to establish some cosmological thermodynamic equations in the context of the cyclical universe as the scenario in which the universe itself is considered like an adiabatic thermodynamical system enclosed in physical volume characterized by periodic reversible transitions. Our model is based on the combination of local and global cosmological time-dependent temperatures {T 0(τ 0),T(τ)} and volumes {V 0(τ 0),V(τ)} instead of the critical temperature T c and volume V c; and the infinitesimal relative variations {dT/T,dV/V}, which are mainly due to the cosmological chaotic fluctuations that are generally ignored in certain oscillating models. By taking into account all these factors, certain equations in the form of d ℓ/ℓ=±η d τ/τ H have been established and from them we derive some others to provide a mechanism that is responsible for the thermodynamic evolution of the cyclical universe.  相似文献   

6.
We have studied the structure of hot accretion flow bathed in a general large-scale magnetic field. We have considered magnetic parameters , where are the Alfvén sound speeds in three direction of cylindrical coordinate (r,φ,z). The dominant mechanism of energy dissipation is assumed to be the magnetic diffusivity due to turbulence and viscosity in the accretion flow. Also, we adopt a more realistic model for kinematic viscosity (ν=αc s H), with both c s and H as a function of magnetic field. As a result in our model, the kinematic viscosity and magnetic diffusivity (η=η 0 c s H) are not constant. In order to solve the integrated equations that govern the behavior of the accretion flow, a self-similar method is used. It is found that the existence of magnetic resistivity will increase the radial infall velocity as well as sound speed and vertical thickness of the disk. However the rotational velocity of the disk decreases by the increase of magnetic resistivity. Moreover, we study the effect of three components of global magnetic field on the structure of the disk. We found out that the radial velocity and sound speed are Sub-Keplerian for all values of magnetic field parameters, but the rotational velocity can be Super-Keplerian by the increase of toroidal magnetic field. Also, Our numerical results show that all components of magnetic field can be important and have a considerable effect on velocities and vertical thickness of the disk.  相似文献   

7.
A most general exact solution to the Einstein-Maxwell equations for static charged perfect fluid is sought in terms of pressure. Subsequently, metrics (e λ and e υ ), matter density and electric intensity are expressible in terms of pressure. Consequently, Pressure is found to be an invertible arbitrary function of ω(=c 1+c 2 r 2), where c 1 and c 2(≠0) are arbitrary constants, and r is the radius of star, i.e. p=p(ω). We present a general solution for charged pressure fluid in terms for ω. We list and discuss some old and new solutions which fall in this category.  相似文献   

8.
The effect of a prevalent magnetic field on static and uniformly rotating self-gravitating cylinders of infinite length is examined. The magnetic field is assumed to consist ofH andH z components, which are taken to be functions of the radial coordinate alone. A variety of magnetic-field configurations are shown to be admissible solutions of equations of motion, from which some feasible cases are presented. A particular magnetic-field configuration having bothH andH z components is studied in detail. The configuration is such that the assumption of a polytropic equation of state reduces the equation governing the density function to a non-homogeneous cylindrical analogue of the Lane-Emden equation for spherical polytropes. The homogeneous case is also studied and shows interesting magnetic-field patterns.  相似文献   

9.
A new class of linear ordinary differential equations with periodic coefficients is found which can be transformed to the Gauss hypergeometric equation, and therefore the monodromy matrices are computable explicitly. These equations appear as the variational equations around a straight-line solution in Hamiltonian systems of the form H = T(p) + V(q), where T(p) and V(q) are homogeneous functions of p and q, respectively.  相似文献   

10.
In this paper, the problem of stationary MHD flow for a rotating toroidal plasma is investigated by assuming that the entropy is a surface quantity. Then, the system of ideal MHD equations is reduced to a single second-order elliptic partial differential equation known as the modified Grad-Shafranov (or Maschke-Perrin) equation. Under the assumption that both the function,P s andf 2 are quadratic polynomials of the flux function, a class of semi-analytical solutions is obtained for a plasma contained in a perfectly conducting toroidal boundary with a rectangular cross section. The flux function, poloidal current and the generalized pressure are obtained and discussed for relevant values of the parameters.  相似文献   

11.
The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if =f y /f x satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.  相似文献   

12.
In this work we propose cyclical reversible transitions as the scenario in which the universe evolves, through a series consisting of reversible expansion, temporary stability, and contraction. Our model is based on the comparison between local and global time-dependent densities {ρ 0(τ 0),ρ(τ)} instead of the critical density ρ c, local and global time-dependent Hubble parameters {H 0(τ 0),H(τ)}, and the variations {Δρ(τ),ΔH(τ)} due to cosmological chaotic fluctuations, which are generally ignored in certain oscillating models. By taking into account all these factors, a rate equation in the form of (H 0/H)2 ∝(ρ 0/ρ) has been established, and from it we derive some others, to provide a mechanism that is responsible for the cyclical reversible transitions. Also, the problems of singularities, black hole overproduction, and the second law of thermodynamics arising in oscillating universe models are conceptually resolved.  相似文献   

13.
A rigorous theoretical investigation of nonlinear electron-acoustic (EA) waves in a plasma system (containing cold electrons, hot electrons obeying a Boltzmann distribution, and hot ions obeying a nonthermal distribution) is studied by the reductive perturbation method. The modified Gardner (MG) equation is derived and numerically solved. It has been found that the basic characteristics of the EA Gardner solitons (GSs), which are shown to exist for α around its critical value α c [where α is the nonthermal parameter, α c is the value of α corresponding to the vanishing of the nonlinear coefficient of the Korteweg-de Vries (K-dV) equation, e.g. α c ≃0.31 for μ=n h0/n i0=0.5, σ=T h /T i =10, n h0, n i0 are, respectively, hot electron and nonthermal ion number densities at equilibrium, T h (T i ) is the hot electron (ion) temperature], are different from those of the K-dV solitons, which do not exist for α around α c , and mixed K-dV solitons, which are valid around α∼α c , but do not have any corresponding double layers (DLs) solution. The parametric regimes for the existence of the DLs, which are found to be associated with positive potential, are obtained. The present investigations can be observed in various space plasma environments (viz. the geomagnetic tail, the auroral regions, the cusp of the terrestrial magnetosphere, etc.).  相似文献   

14.
The stellar equilibrium equations for given surface pressureP * and temperatureT *, and in the absence of convection, are translated into a nonlinear integral equation, in which the radiusR of the star enters as an eigenvalue. We show that under broad mathematical assumptions on the constitutive equations (equation of state, opacity and energy generation) a global existence and uniqueness property can be formulated. If a valueP M is selected, which restricts the allowed pressure and temperature range |P(r)–P *|+E|T(r)–T *P M (E, arbitrary constant of dimensions of a pressure over temperature), thenat least one solutionP(r),T(r) exists in the pressure-temperature range chosen, for anyR<R E . This solution isunique forR<R c .R E andR c are expressed in terms of the constitutive equations, and of the pressure-temperature range adopted. A physical argument in favour of the stability of this solution is presented.  相似文献   

15.
An exact solution of the transport equation in radiative transfer for an axially symmetric Rayleigh scattering problem in semi-infinite planetary atmosphere both for emergent intensity and intensity at any optical depth has been derived with the help of the Laplace transform and the Wiener-Hopf technique, and by use of the constancy of net flux. Chandrasekhar's results for emergent intensity have been verified. New expressions for theH l andH r functions have been obtained.  相似文献   

16.
We compute the signs of two different current helicity parameters (i.e., &amp;#x03B1;best andH c) for 87 active regions during the rise of cycle 23 The results indicate that 59% of the active regions in the northern hemisphere have negative &amp;#x03B1;best and 65% in the southern hemisphere have positive. This is consistent with that of the cycle 22. However, the helicity parameterH cshows a weaker opposite hemispheric preference in the new solar cycle. Possible reasons are discussed.  相似文献   

17.
The Gibbs thermodynamic potential of a proton vortex interacting with the normal core of a neutron vortex of radius r << λ (λ is the penetration depth) that is parallel to it and has an outer boundary of radius b is calculated. It is shown that, under this assumption, the capture of only one vortex by the core is energetically favorable. The force acting on the proton vortex owing to the entrained current is found and it is always directed toward the core. The corresponding force for a proton antivortex is directed toward the outer boundary of the neutron vortex. The Ginzburg-Landau equation is solved for a vortex-antivortex system and its Gibbs function is calculated. It is shown that at large distances from the core, vortex-antivortex pairs can form because of fluctuations. Acted on by the entrainment current, the antivortex moves outward, while the vortex stays inside the neutron vortex. It is shown that the best conditions for fluctuational pair production, followed by separation, exist near the outer boundary. It is shown that new proton vortices can develop only in a region where the entrainment magnetic field strength H (ρ) > HC1 (HC1 is the lower critical field). __________ Translated from Astrofizika, Vol. 51, No. 1, pp. 139–149 (February 2008).  相似文献   

18.
We consider a spherical, solid planet surrounded by a thin layer of an incompressible, inviscid fluid. The planet rotates with constant angular velocity.Within the constraints of the geostrophic approximation of hydrodynamics, we determine the equation that governs the motion of a vortex tube within this rotating ocean. This vorticity equation turns out to be a nonlinear partial differential equation of the third order for the stream function of the motion.We next examine the existence of particular solutions to the vorticity equation that represent travelling waves of permanent form but decaying at infinity. A particular solution is obtained in terms of I 1 and k 1, the modified Bessel functions of order one.The question whether these localized vortices that move like solitary waves could even be solitons depends on their behavior during and after collision with each other and has not yet been resolved.Retired, U.S. Naval Research Laboratory, Washington, D.C., U.S.A.  相似文献   

19.
The behavior of quantum dust ion-acoustic (QDIA) shocks in a plasma including inertialess quantum electrons and positrons, classical cold ions and stationary negative dust grains are studied, using a quantum hydrodynamic model (QHD). The effect of dissipation due to the viscosity of ions is taken into account. The propagation of small but finite amplitude QDIA shocks is governed by the Kortoweg-de Vries-Burgers (KdVB) equation. The existence regions of oscillatory and monotonic shocks will depend on the quantum diffraction parameter (H) and dust density (d) as well as dissipation parameter (η 0). The effect of plasma parameters (d,H,η 0), on these structures is investigated. Results indicate that the thickness and height of monotonic shocks; oscillation amplitude of the oscillatory shock wave and it’s wavelength effectively are affected by these parameters. Additionally, the possibility of propagation of both compressive and rarefactive shocks is investigated. It is found that depending on some critical value of dust density (d c ), which is a function of H, compressive and rarefactive shock waves can’t propagate in model plasma. The present theory is applicable to analyze the formation of nonlinear structures at quantum scales in dense astrophysical objects.  相似文献   

20.
As a possible extension of recent work we study the following version of the inverse problem in dynamics: Given a two-parametric familyf(x, y, b)=c of plane curves, find an autonomous dynamical system for which these curves are orbits.We derive a new linear partial differential equation of the first order for the force componentsX(x, y) andY(x, y) corresponding to the given family. With the aid of this equation we find that, depending on the given functionf, the problem may or may not have a solution. Based on given criteria, we present a full classification of the various cases which may arise.  相似文献   

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