An inverse problem in a three-dimensional radiative transfer

In a series of papers (cf. Bellmanet al., 1965a, b; Kagiwadaet al., 1975), an estimation of optical properties of turbid media has been made, in the least-squared sense, with the aid of quasi-linearization and invariant imbedding. Recently, an extension of the above procedure to the three-dimensional case with horizontally inhomogeneous albedo of the underlying surface has been attempted (Ueno, 1982). From computational aspects the numerical evaluation is not so easy, even by means of high-speed electronic computers. In the present paper it is shown that the latin-square algorithm is a useful estimate for the least-squares inference of the optical properties of turbid media bounded by an inhomogeneous reflecting surface.     

Solution of an inverse problem of the dynamics of a particle

The three-dimensional inverse problem of particle dynamics is studied here. The potentialU and the corresponding energyh are determined by the given family of possible trajectories. The classification of the solutions due to the geometry of the given family is obtained.     

Explicit solutions of the three-dimensional inverse problem of dynamics,using the frenet reference frame

Given a two-parameter of three-dimensional orbits, we construct the unit tangent vector, the normal and the binormal which define the Frenet reference frame. In this frame, by writing that the force is conservative, we explicitly obtain the potential as a function of the energy along the trajectories and of its derivatives.Observatoire de Besançon     

On the two-dimensional inverse problem of dynamics

The authors extend the deduction of the equations satisfied by the force fields from inertial to rotating frames, when the curves of a certain family are known to be solutions for the equations of motion. Then Drǎmbâ's equation is obtained as a consequence of this result. The works of Hadamard and Moiseev are proved to be closely related to the inverse problem of dynamics.     

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 symplectic character of the three-dimensional magnetic-binary problem

The scale-covariant theory of gravitation, which was proposed by Canutoet al. (1977), assumes that gravitational and atomic units are related by a scalar function of space-time. The cosmology based on this theory can explain very well the 3K radiation, as well as Hubble's redshift law. However, Falik (1979) derived a result from the scale-covariant cosmology which was in conflict with the observed helium abundance. Contrary to Falik, we show in this paper that if we choose a gauge functionβ～t ^α and α satisfies the condition ?1/3<α<1, the scale-covariant cosmology will be consistent with the observed cosmic-helium abundance. The value α=1/2 has recently been suggested by Canuto and Goldman (1982) on general grounds.     

Unbiased image reconstruction as an inverse problem

An unbiased method for improving the resolution of astronomical images is presented. The strategy at the core of this method is to establish a linear transformation between the recorded image and an improved image at some desirable resolution. In order to establish this transformation only the actual point spread function and a desired point spread function need be known. No image actually recorded is used in establishing the linear transformation between the recorded and improved image.
This method has a number of advantages over other methods currently in use. It is not iterative, which means it is not necessary to impose any criteria, objective or otherwise, to stop the iterations. The method does not require an artificial separation of the image into 'smooth' and 'point-like' components, and thus is unbiased with respect to the character of structures present in the image. The method produces a linear transformation between the recorded image and the deconvolved image, and therefore the propagation of pixel-by-pixel flux error estimates into the deconvolved image is trivial. It is explicitly constrained to preserve photometry and should be robust against random errors.     

The inverse planetary problem: a numerical treatment

The so-called inverse planetary problem can be stated as follows: given the distances from the centre, masses, and radii of (say) three planets of a planetary system, find the optimum polytropic index, mass, and radius of their star, and also other quantities of interest, which depend either explicitly or implicitly on the foregoing ones (e.g., central and mean density, central and mean pressure, central and mean temperature, etc.). It is hereafter tacitly assumed that the system is opaque with respect to observations concerning periods of planetary otbits; hence, we cannot have any relevant estimates due to the well-known period laws. In the present paper, the inverse planetary problem is treated numerically on the basis of the so-called global polytropic model, developed recently by the first author.     

The bifurcation of periodic orbits of the planar circular restrictedN-body problem to the three-dimensional generalN-body problem

It is proved that the vertical critical orbits of the planar circular restricted three-body problem can be used as starting points for finding periodic orbits of the three-dimensional generalN-body problem. A numerical example is given.     

Zero velocity hypersurfaces for the general three-dimensional three-body problem

In a recent paper by Sergysels with the same title, he describes a problem concerning these surfaces. In this brief commentary, we resolve this problem.     

Zero velocity hypersurfaces for the general three-dimensional three-body problem

The equation of zero velocity surfaces for the general three-body problem can be derived from Sundman's inequality. The geometry of those surfaces was studied by Bozis in the planar case and by Marchal and Saari in the three-dimensional case. More recently, Saari, using a geometrical approach, has found an inequality stronger than Sundman's. Using Bozis' algebraic method, and a rotating frame which does not take into account entirely the rotation of the three-body system, we also find an inequality stronger than Sundman's. The comparison with Saari's inequality is more difficult. However, our result can be expressed in four-dimensional space and the regions where motion is allowed can be seen (numerically) to lie inside those obtained by the use of Sundman's inequality.Agrégé de Faculté.     

The flattening of three-dimensional periodic orbits in the restricted problem

A number of partly known families of symmetric three-dimensional periodic orbits of the restricted three-body (=0.4) problem are numerically continued in both ends until they terminate with orbits in the plane of motion of the primaries. The families of plane symmetric periodic orbits from which they bifurcate are identified and many orbit illustrations are given.     

Integrals of motion in the elliptic three-dimensional restricted three-body problem

Three integrals of motion have been found in the three-dimensional elliptic restricted three-body problem for small eccentricitye of the relative orbit of the primaries and small distancer and eccentricitye of the orbit of the third body around a primary. The integrals are given in the form of formal series in the mass-ratio , the eccentricitiese, e and the coordinates and velocities. These integrals depend periodically on the time.     

The existence of three-dimensional symmetric orbits in the magnetic-binary problem

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.     

Surface fitting of three-dimensional closed solutions of the restricted three-body problem

The orbits of a family of three-dimensional periodic orbits in the restricted problem of three bodies form a surface. In this paper we determine the equation of this surface in the case of the orbits of double symmetry of the family which emanates from the equilibrium pointL ₁. This equation is obtained numerically by a least squares approximation method.     

Separable and partially separable systems in the light of the inverse problem of dynamics

Using a generalization of Joukovsky's formula, we determine three-dimensional families of curves that are orbits only in separable potentials and we note the importance of iso-energetic families of orbits. We also obtain more general families that are orbits of partially separable systems and we examine from this point of view the classical curvilinear coordinate systems.     

Families of symmetric-periodic orbits in the elliptic three-dimensional restricted three-body problem

With an orbit of the three-dimensional circular problem as a starting point, we have calculated families of symmetric-periodic orbits in the three-dimensional elliptic problem with a variation of the mass ratio and the eccentricitye. Afterwards, we have studied their evolution and stability.     

Solution of Lambert's problem for short arcs

Approximation formulas are found for and , wherex(t) satisfies ,x(0)=x ₀,x(1)=x ₁. The results are applied to an example of two-body motion.