Bruns' Theorem: The Proof and Some Generalizations

We give here a proof of Bruns’ Theorem which is both complete and as general as possible: Generalized Bruns’ Theorem.In the Newtonian (n+1)-body problem in ℝ ^p with n≥2 and 1≤p≤n+1, every first integral which is algebraic with respect to positions, linear momenta and time, is an algebraic function of the classical first integrals: the energy, the p(p−1)/2 components of angular momentum and the 2p integrals that come from the uniform linear motion of the center of mass. Bruns’ Theorem only dealt with the Newtonian three-body problem in ℝ³; we have generalized the proof to n+1 bodies in ℝ^p with p≤n+1. The whole proof is much more rigorous than the previous versions (Bruns, Painlevé, Forsyth, Whittaker and Hagiara). Poincaré had picked out a mistake in the proof; we have understood and developed Poincaré’s instructions in order to correct this point (see Subsection 3.1). We have added a new paragraph on time dependence which fills in an up to now unnoticed mistake (see Section 6). We also wrote a complete proof of a relation which was wrongly considered as obvious (see Section 3.3). Lastly, the generalization, obvious in some parts, sometimes needed significant modifications, especially for the case p=1 (see Section 4). This revised version was published online in July 2006 with corrections to the Cover Date.     

Fast computation of complete elliptic integrals and Jacobian elliptic functions

As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K(m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9–19 obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, K(1 − m) and E(1 − m), are first computed by a pair of the approximate polynomials and then transformed to K(m) and E(m) by means of Jacobi’s nome, q, and Legendre’s identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody’s Chebyshev polynomial approximation of Hastings type and Innes’ formulation based on q-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, sn(u|m), cn(u|m), and dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, u, after its domain is reduced to the standard range, 0 ≤ u < K(m)/4, with the help of the new method to compute K(m). The new procedure is 25–70% faster than the methods based on the Gauss transformation such as Bulirsch’s algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of K(m) is not taken into account.     

On the third integral of the galaxy

It is shown here that the third integral of the galaxy, whenever its constant is conserved, defines the same surface as the Hamiltonian, and thus does not constitute anynew integral, but a function of the already known integral of energy. In particular, the third integral and the Hamiltonian are found to possess collinear gradients, in accordance with Poincaré's theorem concerning the characteristic exponents in systems with multiple integrals.     

The haloing effect of the third integral

Whether Contopoulos's galactic system is separable (unlikely) or not (likely), the fact is that there exists a vicinity of the equilibrium in which numerical integration of high accuracy cannot separate the system from its image through Birkhoff's normalization of high order. To all practical purposes, stellar dynamics is then justified in pretending that the model is, in that region, structured by a so-called third integral.     

General integral transform of the exponential integrals

General integral transform of the exponential integralsE _n is considered and will be denoted asB _(k) ⁿ (). Different expressions and the equations satisfied byB _(k) ⁿ are developed. Two-term recurrence formula forB _(k) ⁿ (0) and three-term recurrence formula forB _(k) ⁿ (); 0 will be established for a givenk1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy     

Inverting the zodiacal light brightness integral

Considerations of the geometry appropriate to observations of the zodiacal light made from out of the ecliptic plane yield the general inversion of the brightness integral. The brightness per unit volume of interplanetary space can thus be determined in the immediate neighborhood of the spacecraft in directions confined to a unique viewing plane which depends upon the spacecraft's trajectory. The implementation of this technique guarantees the maximum information content of optical observations made from future deep-space probes including the “Out-of-Ecliptic” mission scheduled for launch in 1983.     

The third integral and the Hamiltonian

It is reiterated that any suggestion of the existence of a third integral is at variance with Poincaré's theorem on the non-existence of such integrals. Even in a purely numerical approach no form of a new integral can be constructed that is valid in every domain of the phase space; and it is devoid of meaning to use as a third integral different forms of functions in various cases.     

Is the third integral a function of the Hamiltonian?

Numerical evidence is presented which indicates that, although the third integral is tangent to the Hamiltonian (energy integral) along some periodic orbits (as has been shown by Goudas), it is not tangent to it along non-periodic orbits; therefore it is not a function of the Hamiltonian. The set of periodic orbits is probably dense in general, but a given form of the third integral is valid in the neighbourhood of a limited number of them; no form of the third integral is valid for all periodic orbits, except in integrable cases.     

Extragalactic integral field spectroscopy on the Gemini telescopes

We have been undertaking a programme on the Gemini 8‐m telescopes to demonstrate the power of integral field spectroscopy, using the optical GMOS spectrograph, and the new CIRPASS instrument in the near‐infrared. Here we present some preliminary results from 3D spectroscopy of extra‐galactic objects, mapping the emission lines in a 3CR radio galaxy and in a gravitationally lensed arc, exploring dark matter sub‐structure through observations of an Einstein Cross gravitational lens, and the star formation time‐scales of young massive clusters in the starburst galaxy NGC 1140. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the energy integral for first post-Newtonian approximation

The post-Newtonian approximation for general relativity is widely adopted by the geodesy and astronomy communities. It has been successfully exploited for the inclusion of relativistic effects in practically all geodetic applications and techniques such as satellite/lunar laser ranging and very long baseline interferometry. Presently, the levels of accuracy required in geodetic techniques require that reference frames, planetary and satellite orbits and signal propagation be treated within the post-Newtonian regime. For arbitrary scalar W and vector gravitational potentials \(W^j (j=1,2,3)\), we present a novel derivation of the energy associated with a test particle in the post-Newtonian regime. The integral so obtained appears not to have been given previously in the literature and is deduced through algebraic manipulation on seeking a Jacobi-like integral associated with the standard post-Newtonian equations of motion. The new integral is independently verified through a variational formulation using the post-Newtonian metric components and is subsequently verified by numerical integration of the post-Newtonian equations of motion.     

Short-period intensity fluctuations of integral sunlight

An attempt has been made to detect short-period solar luminosity fluctuations in the vicinity of 5 min, analogous to the observed velocity oscillation. Using silicon photodiodes to monitor integral sunlight, an upper limit for the amplitude of the intensity fluctuations of 3 x 10^-5 rms was found.Proceedings of the 14th ESLAB Symposium on Physics of Solar Variations, 16–19 September 1980, Scheveningen, The Netherlands.     

The second integral of the restricted problem in regularized elements

A perturbation series integral for the restricted problem of three bodies is derived by use of a new set of canonical elements for the regularized two-body problem. These elements are similar to theKS elements of Stiefel and Scheifele, but they contain small parameters other than the semimajor axis. The variable analogous to the longitude of perihelion not only remains well defined as the orbit approaches a circle, but also it can be used as a second small parameter. Regularized elements permit canonical use of the eccentric anomaly as independent variable, but most of the major benefits of regularization in the two-body problem do not carry over to perturbation theory.     

The integral equation approach to the study of interplanetary dust

Some considerations about the zodiacal light brightness integral from the stand point of the theory of integral equations are made. It is shown that for observation directions confined to a plane perpendicular to the ecliptic and passing through the Sun, the Z.L. brightness integral can be formally considered as a first kind integral equation of Volterra type (V.I.E.). In a second step, this equation is transformed into a V.I.E. of the second kind, from which, and under certain assumptions, the spatial distribution of dust out of the ecliptic is obtained.     

Application of the integral variation method to satellite orbit prediction

The Integral Variation (IV) method is a technique to generate an approximate solution to initial value problems involving systems of first-order ordinary differential equations. The technique makes use of generalized Fourier expansions in terms of shifted orthogonal polynomials. The IV method is briefly described and then applied to the problem of near Earth satellite orbit prediction. In particular, we will solve the Lagrange planetary equations including the first three zonal harmonics and drag. This is a highly nonlinear system of six coupled first-order differential equations. Comparison with direct numerical integration shows that the IV method indeed provides accurate analytical approximations to the orbit prediction problem.Advanced Systems Studies; Bldg. 254EElectro-Optical Systems Laboratory; Bldg. 201.     

A reduction of the magnetic-binary problem by means of the energy integral

The author's aim in this article has been to present the magnetic-binary problem in a new form different from the usual, by using a process of reduction of a dynamical problem to another one with fewer degrees of freedom.     

Light scattering calculations by the Fredholm integral equation method

The Fredholm integral equation method (FIM), originally introduced by Holtet al. to solve the light scattering problem for ellipsoidal particles, is reinvestigated by taking into account a recent great progress in numerical computers. A numerical code optimized for vector-processing computers is developed, and is applied to the light scattering by spherical and spheroidal particles. The results for these particles are compared with those by the Mie theory and by Asano and Yamamoto, respectively, and it is confirmed that the agreement with both of them is satisfactory. Sample calculations are also performed for the oblique incidence, in which the direction of incidence is not parallel nor perpendicular to the symmetry axis of the particle. No difficulties in the computation are found compared with the calculations for the parallel or perpendicular incidence. We study the efficiency factor for polarization (Q _pol) in general direction of incidence for spheroidal particles, and discuss the deviation from the Rayleigh approximation.     

A three-dimensional equation for the inversion of the zodiacal light brightness integral

A three-dimensional equation for deriving the scattering properties per unit volume of interplanetary space is obtained. The equation is related to the zodiacal light intensity observed from a photometer set in some arbitrary position relative to the zodiacal light symmetry plane.     

A simplification of the Poisson integral in the theory of spiral density waves

We give here a method of simplifying the evaluation of the Poisson integral in the spiral density wave theory, which will save computer time in the evaluation of the kernal function by a factor of 84. A systematic check of the method has been made for a series of asymptotic density — gravitational potential relations.     

The surface integral approach to Radarclinometry

Because radarclinometry is fundamentally describable in terms of a nonlinear, first-order, partial differential equation, one expects that it can, in principle, be carried out by direct deterministic integration beginning at a given threshold profile along the azimuthal coordinate. Such a boundary condition could be provided by the altimetry profile obtained on a preceding or succeeding orbital revolution of the radar-bearing spacecraft. Notwithstanding the mismatched resolutions of the radar altimeter and the radar imaging system as planned for the Megallan mission to Venus, there are fundamental considerations, not involving system noise, that influence the possibility of success of this approach. From the topographic map of the Lake Champlain West quadrangle in the Adirondack Mountains of the U.S., a radar image is synthesized. Radarclinometry, in surface integral form, recaptures the topographic map when the applicable radar reflectance function is weakly variable over the range of application, but it diverges beyond a certain point for nominally variable reflectance functions. The effect can be understood by using results from the shape-from-shading literature. (This literature is produced by a group within the artificial intelligence community who have been independently attacking, for all practical purposes, photoclinometry, except that they have not given primacy to images of terrain.) The ubiquity of the instability suggests that the value of the surface integral approach is much in doubt.