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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 new analysis of radar observations of inner planets for the time span 1964–1989 is described. The residuals show that Mercury topography is an important source of systematic errors which have not been taken into account up to now. The longitudinal and latitudinal variations of heights of Mercury surface were found and an approximate map of equatorial zone |?|≤120° was constructed. Including three values characterizing global nonsphericity of Mercury surface into the set of parameters under determination allowed to improve essentially all estimates. In particular, the variability of the gravitational constantG was evaluated: $$\dot G/G = (0.47 \pm 0.47) \times 10^{ - 11} yr^{ - 1} $$ . The correction to Mercury perihelion motion: $$\Delta \dot \pi = - 0''.017 \pm 0''.052 cy^{ - 1} $$ and linear combination of the parameters of PPN formalism: $$\upsilon = (2 + 2\gamma - \beta )/3 = 0.9995 \pm 0.0013$$ were determined; they are in a good agreement with General Relativity predictions. The obtained values Δ.π and ν correspond to the negligible solar oblateness, the estimate of solar quadrupole moment being: $$J_2 = ( - 0.13 \pm 0.41) \times 10^{ - 6} $$ .  相似文献   

3.
Some useful results and remodelled representations ofH-functions corresponding to the dispersion function $$T\left( z \right) = 1 - 2z^2 \sum\limits_1^n {\int_0^{\lambda r} {Y_r } \left( x \right){\text{d}}x/\left( {z^2 - x^2 } \right)} $$ are derived, suitable to the case of a multiplying medium characterized by $$\gamma _0 = \sum\limits_1^n {\int_0^{\lambda r} {Y_r } \left( x \right){\text{d}}x > \tfrac{1}{2} \Rightarrow \xi = 1 - 2\gamma _0< 0} $$   相似文献   

4.
The Ideal Resonance Problem, defined by the Hamiltonian $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x,\mu \ll 1,$$ has been solved in Garfinkelet al. (1971). As a perturbed simple pendulum, this solution furnishes a convenient and accurate reference orbit for the study of resonance. In order to preserve the penduloid character of the motion, the solution is subject to thenormality condition, which boundsAB" andB' away from zero indeep and inshallow resonance, respectively. For a first-order solution, the paper derives the normality condition in the form $$pi \leqslant max(|\alpha /\alpha _1 |,|\alpha /\alpha _1 |^{2i} ),i = 1,2.$$ Herep i are known functions of the constant ‘mean element’y', α is the resonance parameter defined by $$\alpha \equiv - {\rm B}'/|4AB\prime \prime |^{1/2} \mu ,$$ and $$\alpha _1 \equiv \mu ^{ - 1/2}$$ defines the conventionaldemarcation point separating the deep and the shallow resonance regions. The results are applied to the problem of the critical inclination of a satellite of an oblate planet. There the normality condition takes the form $$\Lambda _1 (\lambda ) \leqslant e \leqslant \Lambda _2 (\lambda )if|i - tan^{ - 1} 2| \leqslant \lambda e/2(1 + e)$$ withΛ 1, andΛ 2 known functions of λ, defined by $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$   相似文献   

5.
In the now classical Lindblad-Lin density-wave theory, the linearization of the collisionless Boltzmann equation is made by assuming the potential functionU expressed in the formU=U 0 + \(\tilde U\) +... WhereU 0 is the background axisymmetric potential and \(\tilde U<< U_0 \) . Then the corresponding density distribution is \(\rho = \rho _0 + \tilde \rho (\tilde \rho<< \rho _0 )\) and the linearized equation connecting \(\tilde U\) and the component \(\tilde f\) of the distribution function is given by $$\frac{{\partial \tilde f}}{{\partial t}} + \upsilon \frac{{\partial \tilde f}}{{\partial x}} - \frac{{\partial U_0 }}{{\partial x}} \cdot \frac{{\partial \tilde f}}{{\partial \upsilon }} = \frac{{\partial \tilde U}}{{\partial x}}\frac{{\partial f_0 }}{{\partial \upsilon }}.$$ One looks for spiral self-consistent solutions which also satisfy Poisson's equation $$\nabla ^2 \tilde U = 4\pi G\tilde \rho = 4\pi G\int {\tilde f d\upsilon .} $$ Lin and Shu (1964) have shown that such solutions exist in special cases. In the present work, we adopt anopposite proceeding. Poisson's equation contains two unknown quantities \(\tilde U\) and \(\tilde \rho \) . It could be completelysolved if a second independent equation connecting \(\tilde U\) and \(\tilde \rho \) was known. Such an equation is hopelesslyobtained by direct observational means; the only way is to postulate it in a mathematical form. In a previouswork, Louise (1981) has shown that Poisson's equation accounted for distances of planets in the solar system(following to the Titius-Bode's law revised by Balsano and Hughes (1979)) if the following relation wasassumed $$\rho ^2 = k\frac{{\tilde U}}{{r^2 }} (k = cte).$$ We now postulate again this relation in order to solve Poisson's equation. Then, $$\nabla ^2 \tilde U - \frac{{\alpha ^2 }}{{r^2 }}\tilde U = 0, (\alpha ^2 = 4\pi Gk).$$ The solution is found in a classical way to be of the form $$\tilde U = cte J_v (pr)e^{ - pz} e^{jn\theta } $$ wheren = integer,p =cte andJ v (pr) = Bessel function with indexv (v 2 =n 2 + α2). By use of the Hankel function instead ofJ v (pr) for large values ofr, the spiral structure is found to be given by $$\tilde U = cte e^{ - pz} e^{j[\Phi _v (r) + n\theta ]} , \Phi _v (r) = pr - \pi /2(v + \tfrac{1}{2}).$$ For small values ofr, \(\tilde U\) = 0: the center of a galaxy is not affected by the density wave which is onlyresponsible of the spiral structure. For various values ofp,n andv, other forms of galaxies can be taken into account: Ring, barred and spiral-barred shapes etc. In order to generalize previous calculations, we further postulateρ 0 =kU 0/r 2, leading to Poisson'sequation which accounts for the disc population $$\nabla ^2 U_0 - \frac{{\alpha ^2 }}{{r^2 }}U_0 = 0.$$ AsU 0 is assumed axisymmetrical, the obvious solution is of the form $$U_0 = \frac{{cte}}{{r^v }}e^{ - pz} , \rho _0 = \frac{{cte}}{{r^{2 + v} }}e^{ - pz} .$$ Finally, Poisson's equation is completely solvable under the assumptionρ =k(U/r 2. The general solution,valid for both disc and spiral arm populations, becomes $$U = cte e^{ - pz} \left\{ {r^{ - v} + } \right.\left. {cte e^{j[\Phi _v (r) + n\theta ]} } \right\},$$ The density distribution along the O z axis is supported by Burstein's (1979) observations.  相似文献   

6.
For an autonomous, conservative, two degree-of-freedom dynamical system, vorticity (the curl of velocity) is constant along the orbit if the velocity field is divergence-free such that: $$u\left( {x, v} \right) - \psi _y , v\left( {x, y} \right) = - \psi _x .$$ Isovortical orbits in configuration space are level curves of a scalar autonomous function Ψ (x, v) satisfying a second-order, non-linear partial differential equation of the Monge-Ampere type: $$2\left( {\psi _{xx} \psi _{yy} - \psi _{xy}^2 } \right) + U_{xx} + U_{yy} = 0,$$ where U(x. y) is the autonomous potential function. The solution Soc the time variable is reduced to a quadrature following determinatio of Ψ. Self-similar solutions of the Monge-Ampere equation under Birkhoff's one-parameter transformation group are derived for homogeneous (power-law) potential functions. It is shown that Keplerian orbits belong to the class of planar isovortical flows.  相似文献   

7.
This short article supplements a recent paper by Dr R. Broucke on velocity-related series expansions in the two-body problem. The derivations of the Fourier and Legendre expansions of the functionsF(v), \(\sqrt {F(\upsilon )} \) and \(\sqrt {{1 \mathord{\left/ {\vphantom {1 {F(\upsilon )}}} \right. \kern-0em} {F(\upsilon )}}} \) are given, where $$F(\upsilon ) = (1 - e^2 )/(1 + 2e\cos \upsilon + e^2 ), e< 1$$ In the two-body problem,v is identified with the true anomaly,e the eccentricity andF(v) equals (an/V)2. Some interesting relations involving Legendre polynomials are also noted.  相似文献   

8.
The Ideal Resonance Problem is defined by the Hamiltonian $$F = B(y) + 2\varepsilon A(y) \sin ^2 x,\varepsilon \ll 1.$$ The classical solution of the Problem, expanded in powers of ε, carries the derivativeB′ as a divisor and is, therefore, singular at the zero ofB′, associated with resonance. With α denoting theresonance parameter, defined by $$\alpha \equiv - B'/|4AB''|^{1/2} \mu ,\mu = \varepsilon ^{1/2} ,$$ it is shown here that the classical solution is valid only for $$\alpha ^2 \geqslant 0(1/\mu ).$$ In contrast, the global solution (Garfinkelet al., 1971), expanded in powers ofμ=ε1/2, removes the classical singularity atB′=0, and is valid for all α. It is also shown here that the classical solution is an asymptotic approximation, for largeα 2, of the global solution expanded in powers ofα ?2. This result leads to simplified expressions for resonancewidth and resonantamplification. The two solutions are compared with regard to their general behavior and their accuracy. It is noted that the global solution represents a perturbed simple pendulum, while the classical solution is the limiting case of a pendulum in a state offast circulation.  相似文献   

9.
If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form 1 $$\begin{array}{*{20}c} {F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 ,} & {\mu \ll 1.} \\ \end{array} $$ Herey is the momentum-vectory k withk=1,2?N, x 1 is thecritical argument, andx k fork>1 are theignorable co-ordinates, which have been eliminated from the Hamiltonian. The purpose of this Note is to summarize the first-order solution of the problem defined by (1) as described in a sequence of five recent papers by the author. A basic is the resonance parameter α, defined by 1 $$\alpha \equiv - B'/\left| {4AB''} \right|^{1/2} \mu .$$ The solution isglobal in the sense that it is valid for all values of α2 in the range 1 $$0 \leqslant \alpha ^2 \leqslant \infty ,$$ which embrances thelibration and thecirculation regimes of the co-ordinatex 1, associated with α2 < 1 and α2 > 1, respectively. The solution includes asymptotically the limit α2 → ∞, which corresponds to theclassical solution of the problem, expanded in powers of ε ≡ μ2, and carrying α as a divisor. The classical singularity at α=0, corresponding to an exact commensurability of two frequencies of the motion, has been removed from the global solution by means of the Bohlin expansion in powers of μ = ε1/2. The singularities that commonly arise within the libration region α2 < 1 and on the separatrix α2 = 1 of the phase-plane have been suppressed by means of aregularizing function 1 $$\begin{array}{*{20}c} {\phi \equiv \tfrac{1}{2}(1 + \operatorname{sgn} z)\exp ( - z^{ - 3} ),} & {z \equiv \alpha ^2 } \\ \end{array} - 1,$$ introduced into the new Hamiltonian. The global solution is subject to thenormality condition, which boundsAB″ away from zero indeep resonance, α2 < 1/μ, where the classical solution fails, and which boundsB′ away from zero inshallow resonance, α2 > 1/μ, where the classical solution is valid. Thedemarcation point 1 $$\alpha _ * ^2 \equiv {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-\nulldelimiterspace} \mu }$$ conventionally separates the deep and the shallow resonance regions. The solution appears in parametric form 1 $$\begin{array}{*{20}c} {x_\kappa = x_\kappa (u)} \\ {y_1 = y_1 (u)} \\ {\begin{array}{*{20}c} {y_\kappa = conts,} & {k > 1,} \\ \end{array} } \\ {u = u(t).} \\ \end{array} $$ It involves the standard elliptic integralsu andE((u) of the first and the second kinds, respectively, the Jacobian elliptic functionssn, cn, dn, am, and the Zeta functionZ (u).  相似文献   

10.
The McGehee's study of the triple collision of the 3-body problem is here applied for the stability of an equilibrium. Let us consider the homogeneous Lagrangian: $$L = \frac{{\dot x^2 + \dot y^2 }}{2} + U(x,y)$$ whereU is polynomial, with degreek. We establish a necessary and sufficient condition onU for the stability of \(\omega (x = y = \dot x = \dot y = 0)\) .  相似文献   

11.
The spheroidal harmonics expressions $$\left[ {P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) - P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ and $$\left[ {\eta ^2 P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) + \xi ^2 P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ , have ξ2+η2 as a factor. A method is presented for obtaining for these two expressions the coefficient of ξ2+η2 in the form of a linear combination of terms of the formP 2m 2s (iξ)P 2n 2s (η)e i2sθ. Explicit formulae are exhibited for the casesr=1, 2, 3 and any positive or zero integersk ands. Such identities are useful in gravitational potential theory for ellipsoidal distributions when matching Legendre function expansions are employed.  相似文献   

12.
Exact solutions of Einstein field equations are obtained in the scalar-tensor theories developed by Saez and Ballester (1985) and Lau and Prokhovnik (1986) when the line-element has the form $$ds^2 = \exp \left( {2h} \right)dt^2 - \exp \left( {2A} \right)\left( {dx^2 + dy^2 } \right) - \exp \left( {2B} \right)dz^2 $$ whereh, A andB are functions oft only. The solutions are spatially homogeneous, locally rotationally symmetric and admit a Bianchi I group of motions on hypersurfacest = constant. The dynamical behaviours of these models have also been discussed.  相似文献   

13.
A spherically-symmetric static scalar field in general relativity is considered. The field equations are defined by $$\begin{gathered} R_{ik} = - \mu \varphi _i \varphi _k ,\varphi _i = \frac{{\partial \varphi }}{{\partial x^i }}, \varphi ^i = g^{ik} \varphi _k , \hfill \\ \hfill \\ \end{gathered} $$ where ?=?(r,t) is a scalar field. In the past, the same problem was considered by Bergmann and Leipnik (1957) and Buchdahl (1959) with the assumption that ?=?(r) be independent oft and recently by Wyman (1981) with the assumption ?=?(r, t). The object of this paper is to give explicit results with a different approach and under a more general condition $$\phi _{;i}^i = ( - g)^{ - 1/2} \frac{\partial }{{\partial x^i }}\left[ {( - g)^{1/2} g^{ik} \frac{\partial }{{\partial x^k }}} \right] = - 4\pi ( -g )^{ - 1/2} \rho $$ where ?=?(r, t) is the mass or the charge density of the sources of the field.  相似文献   

14.
Using pseudo-time-dependent models and three different reaction networks, a detailed study of the dominant reaction pathways for the formation of cyanopolyynes and their abundances in TMC-1 is presented. The analysis of the chemical reactions show that for the formation of cyanopolyynes there are two major chemical regimes. First, early times of less than ～104 yrs when ion-molecule reactions are dominant, the main chemical route for the formation of larger cyanopolyynes is $$C_n H^ + \xrightarrow{N}C_n N^ + \xrightarrow{{H_2 }}HC_n N^ + \xrightarrow{{H_2 }}H_2 C_n N^ + \xrightarrow{{e^ - }}HC_n N$$ wheren=5, 7, and 9. Second, at times greater than 104 yrs, when neutral-neutral reactions become dominant, two major reaction routes for the formation of cyanopolyynes are (a), $$HCN\xrightarrow{{C_2 H}}HC_3 N\xrightarrow{{C_2 H}}HC_5 N\xrightarrow{{C_2 H}}HC_7 N\xrightarrow{{C_2 H}}HC_9 N$$ and (b) $$C_n H_2 + CN \to HC_{n + 1} N + H,{\text{ }}n = 4,6, and 8$$ depending on the reaction network used. The results indicate that for route (a) large abundances ofC 2 H (fractional abundances of ～10?7), and for route (b) large abundances ofC 2 H 2 are required in order to reproduce the observed abundances of cyanopolyynes. The calculated abundances of cyanopolyynes show great sensitivity to the value of extinction particularly att?5×105 yrs (i.e. photochemical timescale). The effect of other physical parameters, such as the cosmic-ray ionization abundances are also examined. In general, the model calculations show that the observed abundances of cyanopolyynes can be achieved by pseudo-time-dependent models at late times of several million years.  相似文献   

15.
The Ideal Resonance Problem in its normal form is defined by the Hamiltonian (1) $$F = A (y) + 2B (y) sin^2 x$$ with (2) $$A = 0(1),B = 0(\varepsilon )$$ where ? is a small parameter, andx andy a pair of canonically conjugate variables. A solution to 0(?1/2) has been obtained by Garfinkel (1966) and Jupp (1969). An extension of the solution to 0(?) is now in progress in two papers ([Garfinkel and Williams] and [Hori and Garfinkel]), using the von Zeipel and the Hori-Lie perturbation methods, respectively. In the latter method, the unperturbed motion is that of a simple pendulum. The character of the motion depends on the value of theresonance parameter α, defined by (3) $$\alpha = - A\prime /|4A\prime \prime B\prime |^{1/2} $$ forx=0. We are concerned here withdeep resonance, (4) $$\alpha< \varepsilon ^{ - 1/4} ,$$ where the classical solution with a critical divisor is not admissible. The solution of the perturbed problem would provide a theoretical framework for an attack on a problem of resonance in celestial mechanics, if the latter is reducible to the Ideal form: The process of reduction involves the following steps: (1) the ration 1/n2 of the natural frequencies of the motion generates a sequence. (5) $$n_1 /n_2 \sim \left\{ {Pi/qi} \right\},i = 1, 2 ...$$ of theconvergents of the correspondingcontinued fraction, (2) for a giveni, the class ofresonant terms is defined, and all non-resonant periodic terms are eliminated from the Hamiltonian by a canonical transformation, (3) thedominant resonant term and itscritical argument are calculated, (4) the number of degrees of freedom is reduced by unity by means of a canonical transformation that converts the critical argument into an angular variable of the new Hamiltonian, (5) the resonance parameter α (i) corresponding to the dominant term is then calculated, (6) a search for deep resonant terms is carried out by testing the condition (4) for the function α(i), (7) if there is only one deep resonant term, and if it strongly dominates the remaining periodic terms of the Hamiltonian, the problem is reducible to the Ideal form.  相似文献   

16.
A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz., $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of discrete ordinates originally due to Chandrasekhar.  相似文献   

17.
In this paper, using two methods: LCN'S (Lyapunov characteristic numbers) method and slice cutting method, we study numerically two mappings with odd dimension: $$T_1 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + z_n ,} \\ {y_{n + 1} = y_n + x_{n + 1} , (\bmod 2\pi )} \\ {z_{n + 1} = z_n + A\sin y_{n + 1} ,} \\ \end{array} } \right. T_2 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + y_n + B \sin z_n ,} \\ {y_{n + 1} = y_n + A \sin x_{n + 1} , (\bmod 2\pi ),} \\ {z_{n + 1} = z_n + B \sin y_{n + 1} ,} \\ \end{array} } \right.$$ whereA, B are parameters. For the mappingT 1 the whole region is stochastic; however, we find two-dimensional invariant manifolds for the mappingT 2.  相似文献   

18.
In this paper we discuss a perturbed extension of hyperbolic twist mappings to a 3-dimensional measure-preserving mapping $$\begin{array}{*{20}c} {T:\left\{ {\begin{array}{*{20}c} {x_{n + 1} = s(x_n \cos \varphi _n - y_n \sin \varphi _n ) + A\cos z_n ,} \\ {y_{n + 1} = s^{ - 1} (x_n \sin \varphi _n + y_n \cos \varphi _n ) + B\sin z_n ,} \\ {z_{n + 1} = z_n + C\cos (x_{n + 1} + y_{n + 1} ) + D,(\bmod 2\pi )} \\ \end{array} } \right.} \\ {\varphi _n = (x_n^2 + y_n^2 )^k } \\ \end{array}$$ wheres, k are parameters andA, B, C, D are perturbation parameters. We find that the ordered regions near the fixed point of the hyperbolic twist mapping is destroyed by the perturbed extension more easily than the ones distant from it. The size of the ordered region decreases with increasing perturbation parameters and is insensitive to the parameterD for the same parametersA, B, C.  相似文献   

19.
The projection of an axially symmetric satellite's orbit on a plane perpendicular to the rotation axis (z=const.) is given by the second-order differential equation. $$\frac{{y''}}{{1 + y'^2 }} = \bar \Psi _y - y'\bar \Psi _{x,}$$ where the prime denotes the derivative with respect tox and \(\bar \Psi (x,y)\) is a known function. Two integrability cases have been investigated and it has been shown that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the calssical plane eikonal equation of geometric optics.  相似文献   

20.
The equation of transfer for interlocked multiplets has been solved by Laplace transformation and the Wiener-Hopf technique developed by Dasgupta (1978) considering two nonlinear forms of Planck function: i.e., (a) $$B{\text{ }}_{\text{v}} (T) = B(t) = b_0 + b_1 {\text{ }}e^{ - \alpha t} ,$$ (b) $$B{\text{ }}_{\text{v}} (T) = B(t) = b_0 + b_1 t + b_2 E_2 (t).$$ Solutions obtained by Dasgupta (1978) or by Chandrasekhar (1960) may be obtained from our solutions by dropping the nonlinear terms.  相似文献   

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