共查询到20条相似文献，搜索用时 265 毫秒
1.
This paper is a sequel to an earlier article of the same title. The two formal analytical solutions of the Ideal Resonance Problem developed respectively by Garfinkel and Jupp are here compared, atsecond-order in the appropriate small parameter, with numerical integrations; the second-order circulation solution for Jupp's theory being presented for the first time. It transpires that throughout most of the deep resonance regime the second-mentioned solution provides greater accuracy. In addition, it is demonstrated that the first solution is not appropriate when general initial values of the variables are prescribed. 相似文献
2.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1987,40(2):87-93
This is the last article in a series of the same title. The two formal solutions of the Ideal Resonance Problem, developed respectively by Garfinkel and Jupp, were compared and contrasted in the earlier papers. It was stated there that the principal shortcoming of Jupp's analytical solution was the occurrence of a singularity at the separatrix. The purpose of this contribution is to demonstrate how this singularity may readily be removed. Accordingly, modified solutions are presented for the libration and circulation regions. 相似文献
3.
Martin Lara 《Celestial Mechanics and Dynamical Astronomy》2014,118(3):221-234
A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter. 相似文献
4.
The second-order solution of the Ideal Resonance Problem, obtained by Henrard and Wauthier (1988), is developed further to fourth order applying the same method. The solutions for the critical argument and the momentum are expressed in terms of elementary functions depending on the time variable of the pendulum as independent variable. This variable is related to the original time variable through a Kepler-equation. An explicit solution is given for this equation in terms of elliptic integrals and functions. The fourth-order formal solution is compared with numerical solutions obtained from direct numerical integrations of the equations of motion for two specific Hamiltonians. 相似文献
5.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1972,5(1):8-26
A second-order libration solution of theIdeal Resonance Problem is construeted using a Lie-series perturbation technique. The Ideal Resonance Problem is characterized by the equations $$\begin{gathered} - F = B(x) + 2\mu ^2 A(x)sin^2 y, \hfill \\ \dot x = - Fy,\dot y = Fx, \hfill \\ \end{gathered} $$ together with the property thatB x vanishes for some value ofx. Explicit expressions forx andy are given in terms of the mean elements; and it is shown how the initial-value problem is solved. The solution is primarily intended for the libration region, but it is shown how, by means of a substitution device, the solution can be extended to the deep circulation regime. The method does not, however, admit a solution very close to the separatrix. Formulae for the mean value ofx and the period of libration are furnished. 相似文献
6.
Boris Garfinkel 《Celestial Mechanics and Dynamical Astronomy》1975,12(2):203-214
An Extended Resonance Problem is defined by the Hamiltonian, $$F = B(y) + 2\mu ^2 A(y)[\sin x + \lambda (y)]^2 \mu<< 1,\lambda = O(\mu ).$$ It is noted here that the phase-plane trajectories exhibit adouble libration, enclosing two centers, for the initial conditions of motion satisfying the inequality $$1 - |\lambda |< |\alpha |< 1 + |\lambda |,$$ where α is the usualresonance parameter. A first order solution for the case of double libration is constructed here by a generalization of the procedure previously used in solving the Ideal Resonance Problem with λ=0. The solution furnishes a reference orbit for a Perturbed Ideal Problem if a double libration occurs as a result of perturbations. 相似文献
7.
An infinitestimal contact transformation is proposed to simplify at first order the Hamiltonian representing the attitude of a triaxial rigid body on a Keplerian orbit around a mass point. The simplified problem reduces to the Euler-Poinsot model, but with moments of inertia depending on time through the longitude in orbit. Should the orbit be circular, the moments of inertia would be constant. 相似文献
8.
Marcello Romano 《Celestial Mechanics and Dynamical Astronomy》2008,100(3):181-189
The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of
inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary
initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular,
the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points
on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type,
which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent
hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic
map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found
analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to
the small set of special cases for which an exact solution of the rotational motion of a rigid body exists. 相似文献
9.
Marcello Romano 《Celestial Mechanics and Dynamical Astronomy》2008,101(4):375-390
New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of
inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for
the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular
to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial
angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed
with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case
of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix
in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of
a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of
an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body
with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic
solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion
time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an
exact solution of the rotational motion of a rigid body exists. 相似文献
10.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1973,7(3):347-355
Poincaré formulated a general problem of resonance in the case of a dynamical system which is reducible to one degree of freedom. He introduced the concept of a global solution; in essence, this means that the domain of the solution(s) covers the entire phase plane, comprising regions of libration and circulation. It is the author's opinion that the technique proposed by Poincaré for the construction of a global solution is impractical. Indeed, in §§201 and 211 ofLes méthodes nouvelles de la méchanique céleste, where he describes the passage from shallow resonance to deep resonance, Poincaré asserts an erroneous conclusion. An alternative procedure, which admits secular terms into the determining function and introduces a regularizing function, is outlined. The latter method has been successfully applied to the Ideal Resonance Problem, which is a special case of the more general problem considered by Poincaré, (Garfinkelet al. (1971); Garfinkel (1972). 相似文献
11.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1973,7(1):91-106
The author's previous studies concerning the Ideal Resonance Problem are enlarged upon in this article. The one-degree-of-freedom Hamiltonian system investigated here has the form $$\begin{array}{*{20}c} { - F = B(x) + 2\mu ^2 A(x)\sin ^2 y + \mu ^2 f(x,y),} \\ {\dot x = - F_y ,\dot y = F_x .} \\ \end{array}$$ The canonically conjugate variablesx andy are respectively the momentum and the coordinate, andμ 2 is a small positive constant parameter. The perturbationf is o (A) and is represented by a Fourier series iny. The vanishing of ?B/?x≡B (1) atx=x 0 characterizes the resonant nature of the problem. With a suitable choice of variables, it is shown how a formal solution to this perturbed form of the Ideal Resonance Problem can be constructed, using the method of ‘parallel’ perturbations. Explicit formulae forx andy are obtained, as functions of time, which include the complete first-order contributions from the perturbing functionf. The solution is restricted to the region of deep resonance, but those motions in the neighbourhood of the separatrix are excluded. 相似文献
12.
The restricted problem in the vicinity of the Lagrangian point L4 is studied by finding a convergent binomial expansion of the disturbing function. Using a Hamiltonian formulation in Delaunay variables and removing the short-period terms a resonance problem (already considered by Giacaglia (1970) in an attempt of enlarging the Ideal Resonance) is obtained. It is shown that this extension is reducible to Garfinkel's ideal resonance in the libration region. 相似文献
13.
We describe the application of the implicit midpoint integrator to the problem of attitude dynamics for low-altitude satellites without the use of quaternions. Initially, we consider the satellite to rotate without external torques applied to it. We compare the numerical solution with the exact solution in terms of Jacobi’s elliptic functions. Then, we include the gravity-gradient torque, where the implicit midpoint integrator proves to be a fast, simple and accurate method. Higher-order versions of the implicit midpoint scheme are compared to Gauss–Legendre Runge–Kutta methods in terms of accuracy and processing time. Finally, we investigate the performance of a parameter-adaptive Kalman filter based on the implicit midpoint integrator for the determination of the principal moments of inertia through observations. 相似文献
14.
M. H. C. F. Lacaz 《Celestial Mechanics and Dynamical Astronomy》1985,35(1):9-17
The stability of the equilibrium points and the behavior near the equilibrium points of an Ideal Double Resonance Problem are studied. In the case where the characteristic roots are purely imaginary and such that the stability cannot be decided with linear terms, the nonlinear terms are considered and some theorems of Arnold and of Khazin are used. 相似文献
15.
Boris Garfinkel 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):359-359
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.
R. A. Howland 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):207-211
A new approach to the librational solution of the Ideal Resonance Problem has been devised--one in which a non-canonical transformation is applied to the classical Hamiltonian to bring it to the form of the simple harmonic oscillator. Although the traditional form of the canonical equations of motion no longer holds, a quasi-canonical form is retained in this single-degree-of-freedom system, with the customary equations being multiplied by a non-constant factor. While this makes the resulting system amenable to traditional transformation techniques, it must then be integrated directly. Singularities of the transformation in the circulation region limit application of the method to the librational region of motion.Computer-assisted algebra has been used in all three stages of the solution to fourth order of this problem: using a general-purpose FORTRAN program for the quadratic analytical solution of Hamiltonians in action-angle variables, the initial transformation is carried out by direct substitution and the resulting Hamiltonian transformed to eliminate angular variables. The resulting system of differential equations, requiring the expected elliptic functions as part of their solution, is currently in the process of being integrated using the LISP-based REDUCE software, by programming the required recursive rules for elliptic integration.Basic theory of this approach and the computer implementation of all these techniques is described. Extension to higher order of the solution is also discussed. 相似文献
17.
E. Piña 《Celestial Mechanics and Dynamical Astronomy》1999,74(3):163-174
The motion of three particles, interacting by gravitational forces, is studied in a new coordinate system given by the principal
axes of inertia, as determined by Euler angles, and using the inertia principal moments and an auxiliar angle as coordinates.
The solution to the particular Lagrange case of the three‐body problem is reviewed and solved in these new coordinates.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
The Ideal Resonance Problem, as formulated in 1966 (Paper I), is defined by the Hamiltonian Following the procedure adopted in the construction of a first-orderglobal solution (Papers II, III, and V), we derive a second-order solution from the von Zeipel-Bohlin recursive algorithm of Paper II. The singularities inherent in the Bohlin expansion in powers of μ have been suppressed by means of theregularizing function of Paper III, and the singularities in the coefficients atAB″=0 have been removed by thenormalization technique of Paper V. As a check, it is shown that the global solution includes asymptotically theclassical solution, expanded in powers ofμ 2, and carrying thecritical divisor B′. 相似文献
19.
A. H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1982,26(4):413-422
Whereas the Bohlin-von Zeipel procedure can be used successfully to construct formal solutions to some resonant dynamical systems, it is shown here that a direct Bohlin-Lie series approach seems not to be feasible. The fact that certain terms lose an order of magnitude on differntiation with respect to the momentum variable leads to a situation which precludes an accurate construction of the first-order term in the generating function. A simple remedy to this impasse is suggested, with particular reference to theIdeal Resonance Problem. 相似文献
20.
Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1974,8(4):523-530
In an earlier publication (Jupp, 1972), a solution of the Ideal Resonance Problem is exhibited explicitly in terms of the mean elements; to second order in the small parameter in the case of libration, and to first order in the case of deep circulation. Both representations possess a singularity when the mean modulus of the Jacobi elliptic functions is unity; this corresponds to the separatrix of the phase plane of the dynamical system.It is shown here that, provided particular coefficients associated with the problem satisfy specific relations, the singularity is removed, and the resulting solution is applicable throughout the deep resonance region.The solution is then expressed in terms of general initial conditions. Again, in general, the solution has a singularity associated closely with the limiting motion, and the circulation part of the solution is restricted to deep circulation. It is shown that when the previously-mentioned coefficients satisfy particular constraints, the singularity is removed. In addition, with the same constraints, the deep-circulation solution is applicable throughout the circulation region. It is of interest that these constraints are quite different from those associated with the mean, element formulation. 相似文献