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.     

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.     

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 existence of the third integral for a family of high-velocity stars

A family of high-velocity stars (Energy-19000 km²/sec², angular momentum 1150 kpc km/sec) were computed using Innanen's (1966) mass model of the Galaxy. The stars were set initially on the curve of zero velocity. The inclination diagram indicates a quasi-isolating third integral.     

About the third integral of charged particle motion in strongly curved magnetic fields

Charged particle motion is studied in magnetic fields with an inversion of field direction and a strong curvature in the reversal region, that means field structures which are expected to play a crucial role in energetics and dynamics of space plasma. Investigations are performed in a typical field model. Due to its symmetry and stationarity two of the integrals of motion of the resulting Hamiltonian immediately arise. The question is analyzed whether the system is integrable (this means regular solutions exist) and of what kind the third integral is. Two different third integrals are found, which are valid in different parts of the phase space. Their validity is estimated analytically for particles, crossing the neutral plane (z = 0) -- in addition the action integral I_z previously unusual in plasma applications is verified numerically. Further numerical research establishes that there are no more third integrals -- all remaining parts of the phase space are filled with chaotic solutions. Because in strongly curved field reversals the conservation of the magnetic moment as an integral of motion is shown to be restricted to very small energies or very large pitch angles, the unusual I_z-integral will be an important tool for solution of plasma problems of cosmical current sheets and plasma boundaries.     

The Hamiltonian Dynamics of the Two Gyrostats Problem

The problem of two gyrostats in a central force field is considered. We prove that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system posseses symmetries. Using them we perform the reduction of the number of degrees of freedom. We show that at every stage of the reduction process, equations of motion are Hamiltonian and give explicit forms corresponding to non-canonical Poisson brackets. Finally, we study the case where one of the gyrostats has null gyrostatic momentum and we study the zero and the second order approximation, showing that all equilibria are unstable in the zero order approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

The contact transformation groups of the Extended Hamiltonian System

The 1-parameter transformation groups (otherwise known as infinitesimal transformations) admitted by a system of differential equations are fundamental to the study of its properties. In this paper we first of all consider 1-parameter groups of contact transformations. Then, by generalizing Noether's theorem, we show how they are fundamental to what I call the Extended Hamiltonian System. Finally, this is illustrated by the extendedN-Body problem.

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Resume Les groupes de transformations à 1 paramètre (appelés aussi transformations infinitésimales) admis par un systeme d'équations différentielles sont fondamentaux dans l'étude de ses propriétés. Dans cet article, nous considérons d'abord les groupes à 1 paramètre de transformations de contact. Ensuite, par la généralisation du théorème de Noether, nous montrons qu'ils sont fondamentaux dans l'étude de ce que j'appelle le Système Hamiltonien Etendu. Enfin ceci est illustré par le problème étendu desN-Corps.

     

The Hamiltonian transformation in quadratic lie transforms

The algorithm for Hamiltonian transformation in the quadratic perturbation technique of one of the authors admits of various equivalent forms. Using as a criterion the number of inter-term multiplications required for transformation, however, the amount of effort required to obtain the transformed Hamiltonian is not equivalent among these forms. Each is considered in some detail, and general guidelines for the choice of most efficient algorithm to be used in a given problem are provided. Their utility is demonstrated by application to Duffing's equation.     

Gravitational Waves and the Reparameterization-Invariant Hamiltonian Formalism

A consistent perturbation theory is formulated on the basis of a recently proposed method of reparameterization-invariant Hamiltonian reduction, and a linearized limit in the oscillator approximation of General Relativity is considered. Possible physical consequences of interpreting the nonzero local Hamiltonian as the gravitational field energy are discussed. The physical and geometrical meanings of reduced action variables are investigated. A qualitative conclusion is drawn about the dependence of the distribution of phases of relict gravitational waves at the current epoch on the initial conditions at the time of the Big Bang.     

Asteroid long-periodic perturbations: The second order Hamiltonian

The Hamiltonian of the second order with respect to the disturbing mass, as defined in the higher order-higher degree theory of asteroid secular perturbations by Yuasa (1973), is expressed in the heliocentric, ecliptic coordinate system. Errors found in the original paper with terms coming from the principal part of the disturbing function are removed, and corrected values of the coefficients are computed. The importance of second-order perturbations and the improvement in the accuracy of proper element determination, achieved by using the newly-obtained coefficients, are demonstrated. Finally, a table of the secular frequencies as functions of the semimajor axis is given, and compared with the analogous one by Kozai (1979).     

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.     

The line integral approach to radarclinometry

Radarclinometry, the invention of which has been previously reported, is a technique for deriving a topographic map from a single radar image by using the dependence upon terrain-surface orientation of the integrated signal of an individual image pixel. The radiometric calibration required for precise operation and testing does not yet exist, but the imminence of important applications justifies parallel, rather than serial, development of radarclinometry and radiometrically calibrated radar. The present investigation reports three developmental advances: (1) The solid angle of integration of back-scattered specific intensity constituting a pixel signal is more accurately accounted for in its dependence on surface orientation than in previous work. (2) The local curvature hypothesis, which removes the requirement of a ground-truth profile as a boundary condition and enables the formulation of the theory in terms of a line integral, has been expanded to include the three possibilities of Local Cylindricity, Local Biaxial Ellipsoidal Hyperbolicity, and Least-Squares Local Sphericity. (3) The theory is integrated in the cross-ground-range direction, which is ill-conditioned compared to the ground-range direction, whereas the original formulation was based on enforced isotropy in the two-dimensional power spectrum of the topography. It was found necessary to prohibit the hypothesis of Local Biaxial Ellipsoidal Hyperbolicity in the cross-range stepping, for reasons not completely clear. Variation in the proportioning between curvature assumptions had produced topographic maps that are in good mutual agreement but not realistic in appearance. They are severely banded parallel to the ground-range direction, most especially at small radar zenith angles. Numerical experimentation with the falsification of topography through incorrect decalibration as performed on a Gaussian hill suggests that the banding and its exaggeration at high radar incidence angles could easily be due to our lack of radiometric calibration.     

Expansion of the planetary Hamiltonian function

We expand the planetary Hamiltonian function with its two parts, the principal and the indirect, up to the seventh order in the planetary masses. We adopt the Jacobi-Radau system of origins. The expansiion is valid for any number of planets.     

Stabilization by manipulation of the Hamiltonian

Numerical integration of unstable differential equations should be avoided since a numerical error during thenth step produces erroneous initial values for the next step and thus deteriorates the subsequent integration in an unstable manner. A method is offered to stabilize the equations of motion corresponding to a given HamiltonianH by transformingH into a new HamiltonianH ^* which is equivalent to the Hamiltonian of a harmonic oscillator. In contrast to other methods of stabilization the realm of canonical mechanics is thus not abandoned. Perturbations are discussed and as examples the Keplerian motion and the motion of a gyroscope are presented.     

Integrability of the Yang-Mills Hamiltonian system

This paper considers the integrability of generalized Yang-Mills system with the HamiltonianH _a (p, q)=1/2(p ₁ ² +p ₂ ² +a ₁ q ₁ ² +a ₂ q ₂ ² )+1/4q ₁ ⁴ +1/4a ₃ q ₂ ⁴ + 1/2a ₄ q ₁ ² q ₂ ² . We prove that the system is integrable for the cases: (A)a ₁=a ₂,a ₃=a ₄=1; (b)a ₁=a ₂,a ₃=1,a ₄=3; (C)a ₁=a ₂/4,a ₃=16,a ₄=6. Our main result is the presentation of these integrals. Only for cases A and B does the Yang-Mills Hamiltonian possess the Painlevé property. Therefore the Painlevé test does not take account of the integrability for the case C.     

Development Of The Hamiltonian Function Using The Jacobi System Of Reference

In this paper we describe two different methods to expand the second term of the planetary Hamiltonian function. The Jacobi system of coordinates is adopted leading to a unique evaluation of the Hamiltonian. Previous analytical or semi-analytical planetary theories suffer from the drawback of computing the perturbation function for each planet, which is quite cumbersome. The inclinations of planets are referred to a common fixed plane and the longitudes to a common origin. This is necessary when we deal with n > 2 planets. The treatment is straightforward, and no complexities appear throughout the analysis. This revised version was published online in July 2006 with corrections to the Cover Date.     

The integral Newton's and MacLaurin's theorems in tensor form

The current paper deals with the investigation of the gravitational potential of heterogeneous ellipsoids and its extension to the tensor potential, since little attention has been given to this point in the last century. In this view, both integral Newton's and integral MacLaurin's theorems are formulated in tensor form. The generalization is extended to heterogeneous homeoids and focaloidally striated ellipsoids, respectively. A discontinuity in the tensor potential is found across a homogeneous, infinitely thin focaloid, which vanishes in the spherical limit. The potential‐energy tensors related to focaloidally striated ellipsoids are expressed in integral form, depending on the density profile. All the results are particularized to the spherical limit, for which both Newton's and MacLaurin's theorems hold. With the aim of illustrating the procedure, an explicit calculation of the potential‐energy tensors is outlined in the special case of homogeneous, spherical configurations. Finally, an application is made to the Coma cluster of galaxies.