The perturbed ideal resonance problem

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.     

Global solution of the ideal resonance problem

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 = ε^1/2. 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).     

A theory of integration for the extended Delaunay method

In this paper a method for the integration of the equations of the extended Delaunay method is proposed. It is based on the equations of the characteristic curves associated with the partial differential equation of Delaunay-Poincaré. The use of the method of characteristics changes the partial differential equation for higher order approximations into a system of ordinary differential equations. The independent variable of the equations of the characteristics is used instead of the angular variables of the Jacobian methods and the averaging principle of Hori is applied to solve the equations for higher orders. It is well known that Jacobian methods applied to resonant problems generally lead to the singularity of Poincaré. In the ideal resonance problem, this singularity appears when higher order approximations of the librational motion are considered. The singularity of Poincaré is non-essential and is caused by the choice of the critical arguments as integration variables. The use of the independent variable of the equation of the characteristics in the place of the critical angles eliminates the singularity of Poincaré.     

Ignorable coordinates in the ideal resonance problem

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.     

Normality condition in the ideal resonance problem

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 $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$      

Computer implementation of a new approach to the ideal resonance problem

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.     

A fourth-order solution of the ideal resonance problem

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.     

A second-order global solution of the ideal resonance problem

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′.     

The ideal resonance problem a comparison of two formal solutions I

Garfinkel's solution of the Ideal Resonance problem derived from a Bohlin-von Zeipel procedure, and Jupp's solution, using Poincaré's action and angle variables and an application of Lie series expansions, are compared. Two specific Hamiltonians are chosen for the comparison and both solutions are compared with the numerical solutions obtained from direct integrations of the equations of motion. It is found that in deep resonance the second-mentioned solution is generally more accurate, while in the classical limit the first solution gives excellent agreement with the numerical integrations.This article represents a summary of a much more extensive programme of research, the complete results of which will be published in a future article.     

The ideal resonance problem a comparison of two formal solutions III

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.     

The ideal resonance problem,a comparison of two formal solutions,II

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.     

Application of the stroboscopic method to the stellar three-body problem

The stellar three-body problem has been approached by directly integrating the equations of motion in the orbital elements. The problem is set up in a barycentric chain with the orbits as perturbed ellipses.The integration was performed using the semi-analytical stroboscopic method, which is particularly useful for solving differential systems that depend on several slow variables and one fast, angular variable. Perturbation theory is applied and the solution for a particular order is obtained by way of successive approximations on the fundamental period of the fast variable.     

An observation on the inverse extended problem of Szebehely

Through the use of Jacobi's formulation of the least action principle, differential equations for Szebehely's problem extended to a holonomic system with n degrees of freedom are obtained.

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Sunto Si riottengono le equazioni differenziali relative al problema di Szebehely esteso ad un sistema olonomo ad n gradi di libertà utilizzando il principio di minima azione nella formulazione di Jacobi.

     

The extended phase space formulation of the Vinti problem

The Vinti problem, motion about an oblate spheroid, is formulated using the extended phase space method. The new independent variable, similar to the true anomaly, decouples the radius and latitude equations into two perturbed harmonic oscillators whose solutions toO(J ₂ ⁴ ) are obtained using Lindstedt's method. From these solutions and the solution to the Hamilton-Jacobi equation suitable angle variables, their canonical conjugates and the new Hamiltonian are obtained. The new Hamiltonian, accurate toO(J ₂ ⁴ ) is function of only the momenta.     

Stability in the double resonance problem

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.     

An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem

The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically obtained solutions.     

The structure of the extended phase space of the Sitnikov problem

The extended phase space of the Sitnikov problem is studied by using a stroboscopic map and computing escape times. Comparisons of phase portraits and plots of escape times reveal the intrinsic connection between the geometry of the phase space and the dynamical behaviour of the system. Properties of the phase space are analysed both in the central regular region and far from it. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)