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1.
For the conservative, two degree-of-freedom system with autonomous potential functionV(x,y) in rotating coordinates; $$\dot u - 2n\upsilon = V_x , \dot \upsilon + 2nu = V_y $$ , vorticity (v x -u y ) is constant along the orbit when the relative velocity field is divergence-free such that: $$u(x,y,t) = \psi _y , \upsilon (x,y,t) = - \psi _x $$ . Unlike isoenergetic reduction using the Jacobi, integral and eliminating the time,non-singular reduction from fourth to second-order occurs when (u,v) are determined explicitly as functions of their arguments by solving for ψ (x, y, t). The orbit function ψ satisfies a second-order, non-linear partial differential equation of the Monge Ampere type: $$2(\psi _{xx} \psi _{yy} - \psi _{xy}^2 ) - 2(\psi _{xx} + \psi _{yy} ) + V_{xx} + V_{yy} = 0$$ . Isovortical orbits in the rotating frame arenot level curves of ψ because it contains time explicitly due to coriolis effects. Rather, (x, y) coordinates along the orbit are obtained, from (u, v) either by numerical integration of the kinematic equations, or by partial differentiation of the Legendre transform ? of ψ. In the latter case, ? is shown to satisfy a non-linear, second-order partial differential equation in three independent variables, derived from the Monge-Ampere Equation. Complete reduction to quadrature is possible when space-time symmetries exist, as in the case of central force motion.  相似文献   

2.
The aim of the planar inverse problem of dynamics is: given a monoparametric family of curves f(x, y) = c, find the potential V (x, y) under whose action a material point of unit mass can describe the curves of the family. In this study we look for V in the class of the anisotropic potentials V(x, y) = v(a2x2 + y2), (a=constant). These potentials have been used lately in the search of connections between classical, quantum, and relativistic mechanics. We establish a general condition which must be satisfied by all the families produced by an anisotropic potential. We treat special cases regarding the families (e. g. families traced isoenergetically) and we present certain pertinent examples of compatible pairs of families of curves and anisotropic potentials. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

3.
We consider the following case of the 3D inverse problem of dynamics: Given a spatial two‐parametric family of curves f (x, y, z) = c1, g (x, y, z) = c2, find possibly existing two‐dimension potentials under whose action the curves of the family are trajectories for a unit mass particle. First we establish the conditions which must be fulfilled by the family so that potentials of the form w (y, z) give rise to the curves of the family, and we present some applications. Then we examine briefly the existence of potentials depending on (x, z), respectively (x, y), which are compatible with the given family (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

4.
A well known theorem of relativistic hydrodynamics states that the streamlines of an isentropic perfect fluid are the future-pointing timelike (FPT) curves extremizing the integral J = ∫ S1 S2 fds, where f is the so-called index function and s the proper time on the world line of the fluid particle. The integral is taken over all possible FPT curves with regular representations xi = xi (s) joining the fixed end events E1, E2. The purpose of this note is to show that the streamlines of an adiabatic perfect fluid can likewise be regarded as extremizing curves of the functional J provided the class of admissible curves consists of those FPT curves satisfying the side condition ui∂iS = 0, ui unit 4-velocity and S the specific proper entropy of the fluid, with the first end point fixed and the second being the end point variable. __________ Published in Astrofizika, Vol. 48, No. 4, pp. 641–647 (October–December, 2005).  相似文献   

5.
In the framework of the inverse problem of dynamics, we face the following question with reference to the motion of one material point: Given a region Torb of the xy plane, described by the inequality g (x, y) ≤ c0, are there potentials V = V (x, y) which can produce monoparametric families of orbits f (x, y) = c (also to be found) lying exclusively in the region Torb? As the relevant PDEs are nonlinear, an answer to this question (generally affirmative, but not with assurance) can be given by the procedure of the determination of certain constants specifying the pertinent functions. In this paper we ease the mathematics involved by making certain simplifying assumptions referring to the homogeneity of both the function g (x, y) (describing the boundary of Torb) and of the slope function γ(x, y) = fy/fx (representing the required family f (x, y) = c). We develop the method to treat the so formulated problem and we show that, even under these restrictive assumptions, an affirmative answer is guaranteed provided that two algebraic equations have in common at least one solution (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

6.
Non-periodic orbits of a natural satellite of the Moon are studied, for the case of the circular three-body problem with the method of surface of section. According to this method, each orbit is represented by a point, in the plane x0\.x, which corresponds to y = 0 and \.y > 0 and a fixed energy. Conclusions are deduced from the shape of this curve for probable collisions of the satellite on the lunar surface. This method of surface of section can be used for the study of orbits which collide with the Moon's surface after a large number of revolutions around the Moon and their study would be difficult to explore with other methods.  相似文献   

7.
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   

8.
A two degree-of-freedom, conservative system is reduced to a single degree-of-freedom, kinematic system with Hamiltonian integral under the change of independent variable: $$dt = \zeta dt (\zeta = \upsilon _x - \upsilon _y )$$ where ζ is the curl (or vorticity) of the velocity field with cartesian inertial componentsu(x, y, t) andv(x, y, t). In the autonomous case whenu t=v t=0, orbits are globally represented by the level curves of an autonomous Hamiltonian functionH(x,y) satisfying a second-order quasilinear partial differential equation (Szebehely's Equation): $$2(H + U)\left( {H_{xx} H_y^2 - 2H_{xy} H_x H_y + H_{yy} H_x^2 } \right) + (H_x U_x + H_y U_y )\left( {H_x^2 + H_y^2 } \right) = 0$$ whereU(x, y) is the autonomous potential function. An inversion of dependent and independent variables reduces this equation to a second-order, ordinary differential equation for a function specifying the orbital curve. The true time variable is recovered by evaluating a quadrature. Fundamental differences exist between this approach and Hamilton-Jacobi theory.  相似文献   

9.
For a given family of orbits f(x,y) = c * which can be traced by a material point of unit in an inertial frame it is known that all potentials V(x,y) giving rise to this family satisfy a homogeneous, linear in V(x,y), second order partial differential equation (Bozis,1984). The present paper offers an analogous equation in a synodic system Oxy, rotating with angular velocity . The new equation, which relates the synodic potential function (x,y), = –V(x, y) + % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaaaa!3780!\[\tfrac{1}{2}\]2(x 2 + y 2) to the given family f(x,y) = c *, is again of the second order in (x,y) but nonlinear.As an application, some simple compatible pairs of functions (x,y) and f(x, y) are found, for appropriate values of , by adequately determining coefficients both in and f.  相似文献   

10.
The exact geometry of the Roche curvilinear coordinates (, , ) in which corresponds to the zero-velocity surfaces is investigated numerically in the plane, as well as in the spatial, case for various values of the mass-ratio between the two point-masses (m 1,m 2) constituting a binary system.The geometry of zero-velocity surfaces specified by -values at the Lagrangian points are first discussed by taking their intersections with various planes parallel to thexy-, xz- andyz-planes. The intersection of the zero-velocity surface specified by the -value at the Lagrangian equilateral-triangle pointsL 4,5 with the planex=1/2 discloses two invariable curves passing through the pointsL 4,5 and situated symmetrically with respect to thexy-plane whose form is independent of the mass-ratio.The geometry of the remaining two coordinates (, ) orthogonal to the zero-velocity surfaces is investigated in thexy- andxz-planes from extensive numerical integrations of differential equations generated from the orthogonality relations among the coordinates. The curves (x, y)=constant in thexy-plane are found to be separated into three families by definite envelopes acting as boundaries whose forms depend upon the mass-ratio only: the inner -constant curves associated with the masspointm 1, the inner -constant curves associated with the mass-pointm 2 and the outer -constant curves. All the -constant curves in thexy-plane coalesce at either of the Lagrangian equilateraltriangle pointsL 4,5, except for a limiting case coincident with thex-axis. The curves (x, z)=constant in thexz-plane are also separated by definite envelopes depending upon the mass-ratio into different families: the inner -constant curves associated with the mass-pointm 1, the inner -constant curves associated with the mass-pointm 2 and the outer -constant curves on both sides out of the envelopes. For larger values ofz, the curves =constant tend asymptotically to the line perpendicular to thex-axis and passing through the centre of mass of the system, except for a limiting case coincident with thex-axis. The geometrical aspects of the envelopes for the curves (x, y)=constant in thexy-plane and the curves (x, z)=constant in thexz-plane are also discussed independently.In the three-dimensional space, the Roche coordinates can be conveniently defined in such a way as to correspond to the polar coordinates in the immediate neighbourhood of the origin, and to the cylindrical coordinates at great distances. From numerical integrations of simultaneous differential equations generating spatial curves orthogonal to the zero-velocity surfaces, the surfaces (x, y, z)=constant and the surfaces (x, y, z)=constant are constructed as groups of such spatial curves with common values of some parameters specifying the respective surfaces.On leave of absence from the University of Tokyo as an Honorary Fellow of the Victoria University of Manchester.  相似文献   

11.
It is shown that the equation of motion Du j/Ds = (e/mc 2)F ji u i , a natural generalization to the curved spacetime of the Heaviside-Lorentz law of ponderomotive force, is equivalent to the metric independent and covariant Van Dantzig's equations of motion dx j [jpi] = 0 or L v p i = 0, where p i is the conjugate momentum 4-vector and v a vector determined by the condition p i v i, only with respect to holonomic coordinates. With respect to an anholonomic system, the Heaviside-Lorentz equation is a particular case of the VD equations valid for a privileged class of anholonomic frames, those consisting of orthogonal unit vectors.  相似文献   

12.
We show that the Hénon-Heiles system with Hamiltonian H=\frac12(y12+y22)+\frac12(ax12+bx22)+\frac13dx23+cx12x2{H=\frac12(y_1^2+y_2^2)+\frac12(ax_1^2+bx_2^2)+\frac13dx_2^3+cx_1^2x_2} is integrable in Liouvillian sense (i.e., the existence of an additional first integral) if and only if c = 0; or \frac dc=1, a=b; or \frac dc=6, a, b{\frac dc=1, a=b; {\rm or}\, \frac dc=6, a, b} arbitrary; or \frac dc=16, b=16a{\frac dc=16, b=16a}. Therefore, we get a complete classification of the Hénon-Heiles system in sense of integrability and non-integrability.  相似文献   

13.
In this article we study the conditions for obtaining canonical transformationsy=f(x) of the phase space, wherey(y 1,y 2,...,y 2n ) andx(x 1,x 2,...,x 2m ) in such a way that the number of variables is increased. In particular, this study is applied to the rotational motion in functions of the Eulerian parameters (q 0,q 1,q 2,q 3) and their conjugate momenta (Q 0,Q 1,Q 2,Q 3) or in functions of complex variables (z 1,z 2,z 3,z 4) and their conjugate momenta (Z 1,Z 2,Z 3,Z 4) defined by means of the previous variables. Finally, our article include some properties on the rotational motion of a rigid body moving about a fixed point.  相似文献   

14.
We have investigated the effects of increasing optical depths on spectral lines formed in a rotating and expanding spherical shell. We have assumed a shell whose outer radius is 3 times the inner radius, with the radial optical depths equal to 10, 50, 100, 500. We have employed a constant velocity with no velocity gradients in the shell. The shell is assumed to be rotating with velocities varying as 1/ρ, whereρ is the perpendicular distance from the axis of rotation, implying the conservation of angular momentum. Two expansion (radial) velocities are treated: (1)V = 0 (static case) and (2)V = 10 mean thermal units. The maximum rotational velocities areV rot = 0, 5, 10 and 20. In the shell where there are no radial motions, we obtain symmetric lines with emission in the wings forV rot = 0 and 5 while forV rot ≥ 10 we obtain symmetric absorption lines. In the case of an expanding shell, we obtain lines with central emission.  相似文献   

15.
Unsteady two-dimensional hydromagnetic flow of an electrically conducting viscous incompressible fluid past a semi-infinite porous flat plate with step function change in suction velocity is studied allowing a first order velocity slip at the boundary condition. The solution of the problem is obtained in closed form and the results are discussed with the aid of graphs for various parameters entering in the problem.Notations B intensity of magnetic field - H magnetic field parameter,H=(M+1/4)1/2–1/2 - h rarefaction parameter - L 1 slip coefficient; ;I, mean free path of gas molecules;f, Maxwell's reflection coefficient - M magnetic field parameter - r suction parameter - t time - t dimensionless time - u velocity of the fluid - u dimensionless velocity of the fluid - U velocity of the fluid at infinity - v suction velocity - v 1 suction velocity att<=0 - v 2 suction velocity att>0 - x distance parallel to the plate - y distance normal to the plate - y nondimensional distance normal to the plate - v kinematic viscosity - electric conductivity of the fluid - density of the fluid - shear stress at the wall - nondimensional shear stress at the wall - erf error function - erfc complementary error function  相似文献   

16.
Conditions are found which are satisfied by the coefficients of the expression being a second integral of the motion of an autonomous dynamical system with two degrees of freedom. The coefficientsA, B. , ,E are differentiable functions of the cartesian position coordinatesx, y. The velocity components are denoted by . It is shown that must be constant andB must be of the formB =f(x+y) +g(x-y) wheref, g are arbitrary.Given andB one can always find the remaining coefficientsA, E and also the corresponding potential and second integral. Depending on the specifica case at hand a certain number of arbitrary constants (or arbitrary functions) enter into the potential and the second integral. To each potential (which may be of the separable or nonseparable type in the coordinatesx andy)there corresponds one integral of the above form.  相似文献   

17.
If a dynamical system ofN degrees of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 , \mu<< 1.$$ Herey is the momentum-vectory k withk=1, 2,...,N, andx 1 is thecritical argument. A first-orderglobal solution,x 1(t) andy 1(t), for theactive variables of the problem, has been given in Garfinkelet al. (1971). Sincex k fork>1 are ignorable coordinates, it follows that $$y_\kappa = const., k > 1.$$ The solution is completed here by the construction of the functionsx k(t) fork>1, derivable from the new HamiltonianF′(y′) and the generatorS(x, y′) of the von Zeipel canonical transformation used in the cited paper. The solution is subject to thenormality condition, derived in a previous paper fork=1, and extended here to 2≤k≤N. It is shown that the condition is satisfied in the problem of the critical inclination provided it is satisfied fork=1.  相似文献   

18.
We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m 1 = m 2 = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m 3 < 10−3, placed initially on the z-axis. We begin by finding for the restricted problem (with m 3 = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m 3 increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m 3 ≈ 10−6, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m 3 = 0. Furthermore, as m 3 increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m 3 values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m 3 = 0 case.  相似文献   

19.
We study the existence of three-dimensional symmetric orbits in a magnetic-binary system. We point out that only two kinds of such orbits exist, depending on the orientation of both magnetic momentsM i,i=1, 2; one with respect to the plane,y=0 and one with respect to thex-axis of the rotating-coordinate system.  相似文献   

20.
A quiescent filament was observed near the center of the disk (N5, W5) with the MSDP spectrograph of the 50 cm refractor of the Pic-du-Midi Observatory on June 17, 1986. We focus our study on the statistical moments of the Dopplershift,V 1, and the intensity,I 1, at the center of a chord of the Hα profile (±0.256 Å), versus the minimum intensityI 0. We use a statistical model simulating a numbern max of threads (of optical thicknessτ 0 and source functionS 0), seen over the chromosphere. The threadsj along the same line-of-sighti are identical except for the velocityv j (gaussian distributionv 0,σ v). We search for the best fit between the observed and simulated quantities:V 1,σ (V 1),I 1,σ(I 1), and the histogram of theI 0 values over the field of view. A good fit is obtained with: (a) threads characterized byτ 0 = 0.2,S 0 = 0.06 (unit of the continuum at disk center), mean upward velocityv 0 = 1.7 km s−1 and gaussian-type velocity distributionσ v = 3.5 km s−1. Other possible values ofτ 0 andσ v are discussed; (b) underlying chromosphere deduced from observed quiet Sun (outside the filament) by modifying the chromospheric velocities: additional mean upward velocity 0.7 km s−1, standard deviation reduced by a factorF c ∼ 0.7. The results are discussed in connection with the values deduced from prominence observations.  相似文献   

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