Notes on Hori Method for Canonical Systems

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.     

Equivalence for lie transforms

The Lie transform method used in Perturbation Theory is based upon an intrinsic algorithm for transforming functions or vector fields by a transformation close to the identity. It can thus be viewed as a specialization of methods and results of differential geometry as is shown in the first part of this paper. In a second part we answer some of the questions left open in connection with the equivalence of the algorithms proposed by Hori and Deprit. From a formal point of view, the methods are shown to be equivalent for non-canonical as well as canonical transformations and a formula relating directly the two generating functions (or vector fields) is presented (formula (5.17)). On the other hand, the equivalence is shown to hold also in the ring ofp-differentiable functions.     

On the numerical transformation of variables in perturbation theory

Once the generating function of a Lie-type transformation is known, canonical variables can be transformed numerically by application of a Runge-Kutta type integration method or any other appropriate numerical integration algorithm. The proposed approach avails itself of the fact, that the transformation is defined by a system of differential equations with a small parameter as the independent variable. The integration of such systems arising in the perturbation theories of Hori and Deprit is discussed. The method allows to compute numerical values of periodic perturbations without deriving explicitly the perturbation series. This saving of an algebraic work is achieved at the expense of multiple evaluations of the generator's derivatives.     

The generalized canonical form of Hori's method for non-canonical systems

Any dynamical system can be put in generalized canonical form through the introduction of a set of auxiliary ‘conjugate’ variables or momenta and solved by perturbation theory based on Lie series. The application of Hori's method for generalized canonical system leads to a new canonical transformation — the Mathieu transformation — defined by the solution of the Hori auxiliary system. This new transformation simplifies the algorithm since the inversion of the solution of the Hori auxiliary system is no longer necessary. In this paper, we wish to show some peculiarities of this technique.     

An extended canonical perturbation method

In this investigation, a procedure is described for extending the application of canonical perturbation theories, which have been applied previously to the study of conservative systems only, to the study of non-conservative dynamical systems. The extension is obtained by imbedding then-dimensional non-conservative motion in a 2n-dimensional space can always be specified in canonical form, and, consequently, the motion can be studied by direct application of any canonical perturbation method. The disadvantage of determining a solution to the 2n-dimensional problem instead of the originaln-dimensional problem is minimized if the canonical transformation theory is used to develop the perturbation solution. As examples to illustrate the application of the method, Duffing's equation, the equation for a linear oscillator with cubic damping and the van der Pol equation are solved using the Lie-Hori perturbation algorithm.This research was supported by the Office of Naval Research under Contract N00014-67-a-0126-0013.     

Canonical solution of the two critical argument problem

The solution by Sessin and Ferraz-Mello (Celes. Mech. 32, 307–332) of the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 21 is considerably simplified by the introduction of canonical variables. An analogous canonical transformation simplifies the elliptic restricted problem.     

Resonance in regular variables II: Formal solutions for central and non-central first-order resonance

Application of the method of Hori to the study of motions in the neighbourhood of the origin in the case of a first-order resonance. Discussion of the Lie-series expansions about the origin and of Hori's averaging principles and auxiliary equation, with emphasis on the introduction of the auxiliary parameter t^*.     

Solution of canonical equations which result from elimination of short-periodic terms in second-order planetary theory

We solve the first order non-linear differential equation and we calculate the two quadratures to which are reduced the canonical differential equations resulting from the elimination of the short period terms in a second order planetary theory carried out through Hori's method and slow Delaunay canonical variables when powers of eccentricities and the sines of semi-inclinations which are >3 are neglected and the eccentricity of the disturbing planet is identically equal to zero. The procedure can be extended to the case when the eccentricity of the disturbing planet is not identically equal to zero. In this latter general case, we calculatedthe two quadratures expressing angular slow Delaunay canonical variable ₁ of the disturbed planet and angular slow Delaunay canonical variable ₂ of the disturbing planet in terms of timet.     

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.     

A New Method of Numerical Solution of Nonsteady Problems in the Theory of Radiative Transfer

A new method is proposed for the numerical solution of nonsteady problems in the theory of radiative transfer. In this method, if the solution at some time t (such as the initial time) is known, then by representing the radiation intensity and all time-dependent quantities (level populations, kinetic temperature, etc.) in the form of Taylor series expansions in the vicinity of t, one can, from the transfer equation and the equations accompanying it (population equations, energy-balance equation, etc.), find all derivatives of that solution at the given time from certain recursive equations. From the Taylor series one can then calculate the solution at some later time t + t, and so forth. The method enables one to analyze nonsteady tradiative transfer both in stationary media and in media with characteristics that vary with time in a given way. This method can also be used to solve nonlinear problems, i.e., those in which the radiation field significantly affects the characteristics of the medium. No iterations are used for this: everything comes down to calculations based on recursive equations. Several problems, both linear and nonlinear, are solved as examples.     

Lagrange variational equations from Hori's method for canonical systems

Hori, in his method for canonical systems, introduces a parameter through an auxiliary system of differential equations. The solutions of this system depend on the parameter and constants of integration. In this paper, Lagrange variational equations for the study of the time dependence of this parameter and of these constants are derived. These variational equations determine how the solutions of the auxiliary system will vary when higher order perturbations are considered. A set of Jacobi's canonical variables may be associated to the constants and parameter of the auxiliary system that reduces Lagrange variational equations to a canonical form.     

A note on the integration of the equations of Lie-Deprit's method for unspecified canonical variables

In this note it will be shown that the equations generated by Lie-Deprit's method for unspecified canonical variables could be solved in the same way as Hori did in his method. Here the notations follow those used by Deprit in his paper.     

Third-order Jupiter — Saturn planetary theory

The construction of a third order J-S theory is presented. The Hori theory of planetary perturbations is employed. No Critical J-S terms due to the 2:5 commensurabilities and its multiples exist, when we take into account the periodic terms of order 0, 1, 2 with respect to the eccentricity- inclination. In this case the Lie series transformation degenerates and is meaningless. The J-S equations of motion for secular perturbations are solved when we neglect in our treatment, the Poisson terms of degree > 2 in the Poincaré canonical variables H _u , K _u , P _u Q _u (u = 1, 2). The Jacobi-Radau referential is adopted, and the theory is expressed in terms of the canonical variables of H. Poincaré.Now at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U.S.A.     

Diffuse reflection and transmission in accordance with the phase function 3/4(1+cos2Θ)

We have considered the transport equation for the problem of diffuse reflection and transmission on Rayleigh's phase function and obtained the exact solution of this equation for angular distributions of the intensities diffusely reflected from the surfacet=0 and diffusely transmitted below the surfacet=t ₀ of a finite atmosphere of optical deptht=t ₀ using the Laplace transform and the theory of singular operators. This is an exact method.     

Canonical and phantom scalar fields as an interaction of two perfect fluids

In this article we investigate and develop specific aspects of Friedmann-Robertson-Walker (FRW) scalar field cosmologies related to the interpretation that canonical and phantom scalar field sources may be interpreted as cosmological configurations with a mixture of two interacting barotropic perfect fluids: a matter component ρ ₁(t) with a stiff equation of state (p ₁=ρ ₁), and an “effective vacuum energy” ρ ₂(t) with a cosmological constant equation of state (p ₂=?ρ ₂). An important characteristic of this alternative equivalent formulation in the framework of interacting cosmologies is that it gives, by choosing a suitable form of the interacting term Q, an approach for obtaining exact and numerical solutions. The choice of Q merely determines a specific scalar field with its potential, thus allowing to generate closed, open and flat FRW scalar field cosmologies.     

Holographic dark energy with time depend gravitational constant in the non-flat Ho?ava-Lifshitz cosmology

We study the holographic dark energy on the subject of Hořava-Lifshitz gravity with a time dependent gravitational constant G(t), in the non-flat space-time. We obtain the differential equation that specify the evolution of the dark energy density parameter based on varying gravitational constant. We find out a relation for the state parameter of the dark energy equation of state to low red-shifts which containing varying G corrections in the non-flat space-time.     

Jacobi Dynamics And The N-Body Problem With Variable Masses

An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations. For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Cosmological models with viscous fluid and time- dependent equation of state in Wesson’s theory

Assuming the time-dependent equation of state p=λ(t)ρ, five dimensional cosmological models with viscous fluid for an open universe (k=−1) and flat universe (k=0) are presented. Exact solutions in the context of the rest mass varying theory of gravity proposed by Wesson (Astron. Astrophys. 119, 145, 1983) are obtained. It is found that the phenomenon of isotropisation takes place in this theory, i.e. the mass scale factor A(t) which characterizes the rest mass of a typical particle is evolving with cosmic time just as the spatial scale factor R(t). It is further found that rest mass is approximately constant in the present universe.     

On stationary solutions of some canonical systems of differential equations

In our article (Zhuravlev, 1979) a formal method of constructing conditionally periodic solutions of canonical systems of differential equations with a quick-rotating phase in the case of sharp commensurability was presented. The existence of stationary (or periodic) solutions of an averaged system of differential equations corresponding to the initial system of differential equations is necessary for an effective application of the method for different problems.Evidently, the stationary solutions do not always exist but in numerous papers on stationary solutions (oscillations or motions), the conditions of existence of such solutions are very often not considered at all. Usually a simple assumption is used that the stationary solutions do exist.Otherwise it is well known that Poincaré's theory of periodic solutions (Poincaré, 1892) let one set up conditions of existence of periodic solutions in different systems of differential equations. Particularly, in papers,Mah (1949, 1956), see alsoexmah (1971), the necessary and sufficient conditions of the existence of periodic solutions of (non-canonical) systems of differential equations which are close to arbitrary non-linear systems are given. For canonical autonomous systems of differential equations the conditions of existence of periodic solutions and a method of calculation are presented in the paperMepmah (1952).In our paper another approach is given and the conditions of existence of stationary solutions of canonical systems of differential equations with a quick-rotating phase are proved. For this purpose Delaunay-Zeipel's transformation and Poincaré's small parameter method are used.     

Equilibrium Structures of Differentially Rotating Primary Components of Binary Stars

In this paper a method is proposed for computing the equilibrium structures and various other observable physical parameters of the primary components of stars in binary systems assuming that the primary is more massive than the secondary and is rotating differentially about its axis. Kippenhahn and Thomas averaging approach (1970) is used in a manner earlier used by Mohan, Saxena and Agarwal (1990) to incorporate the rotational and tidal effects in the equations of stellar structure. Explicit expressions for the distortional terms appearing in the stellar structure equations have been obtained by assuming a general law of differential rotation of the typeω² = b ₀+b ₁ s ²+b ₂ s ⁴, where ω is the angular velocity of rotation of a fluid element in the star at a distance s from the axis of rotation, and b ₀, b ₁, b ₂ are suitably chosen numerical constants. The expressions incorporate the effects of differential rotation and tidal distortions up to second order terms. The use of the proposed method has been illustrated by applying it to obtain the structures and observable parameters of certain differentially rotating primary components of the binary stars assuming the primary components to have polytropic structures. This revised version was published online in July 2006 with corrections to the Cover Date.