A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)

FROM the beginning of civilization one need must make itself felt, namely, for some form of calendar. In turn, this gives rise to the primitive study of astronomy, involving some knowledge of the sun and the moon. When the planets are added in the next stage, the astronomer needs an ephemeris or almanac ; the modern Nautical Almanac, with its perfection within narrow limits, is the final outcome. There have been many stages on the road, and some apparent interruptions. But the urge has always been the same, essentially a practical one. The Analytical Foundations of Celestial Mechanics By Aurel Wintner. (Princeton Mathematical Series, No. 5.) Pp. xii + 448. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Analytical Foundations of Celestial Mechanics

In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Transformation of Groups

The treatment of persistent organic pollutants in wastewater from chemical factories by decomposition using advanced oxidation processes (AOPs) has recently been studied. However, the AOPs studied to date suffer from problems such as high running costs for ozone generation and difficulties with operational parameters in the addition of hydrogen peroxide. In this study, a compact AOP reactor that generates ozone by UV irradiation was developed. Ozone was generated using the Chapman method, i.e., by irradiating oxygen with UV light; the use of this method saves energy. The decomposition activity of the designed reactor and the effects of a combination of ozone, UV, and TiO 2 were examined. The results showed that 100% decomposition of 50 mg/dm 3 phenol was reached within 120 min using an O 3 –UV–TiO 2 process, and COD removal reached 100% within 240 min. COD analysis confirmed that a synergistic effect was obtained using ozone, UV, and TiO 2 simultaneously.

Evaluation of advanced oxidation processes (AOP) using O3, UV, and TiO2 for the degradation of phenol in water

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.

/pdf/the-n-body-problem-in-spaces-of-constant-curvature-part-i-1d9112dcdu.pdf

The n-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We analyze the singularities of the equations of motion and several types of singular solutions of the n-body problem in spaces of positive constant curvature. Apart from collisions, the equations encounter noncollision singularities, which occur when two or more bodies are antipodal. This conclusion leads, on the one hand, to hybrid solution singularities for as few as three bodies, whose orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces nonsingularity collisions, characterized by finite velocities and forces at the collision instant.

/pdf/the-n-body-problem-in-spaces-of-constant-curvature-part-ii-54nwfz2byr.pdf

The n -Body Problem in Spaces of Constant Curvature. Part II: Singularities

We generalize the Newtonian n-body problem to spaces of cur- vature � = constant, and study the motion in the 2-dimensional case. For � > 0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite ve- locities and forces at the collision instant. We also point out the existence of several classes of relative equilibria, including the hyperbolic rotations for � 0, hyperbolic relative equilibria to � < 0, and Lagrangian orbits of arbitrary masses to � = 0—results that provide new criteria towards understanding the large-scale geometry of the physical space.

THE n-BODY PROBLEM IN SPACES OF CONSTANT CURVATURE

https://www.math.uvic.ca/faculty/diacu/JDE-2011.pdf

Homographic solutions of the curved 3-body problem

We consider the gravitational motion of n point particles with masses m1,m2, . . . ,mn > 0 on surfaces of constant positive Gaussian curvature. Using stereographic projection, we express the equations of motion defined on the two-dimensional sphere of radius R in terms of the intrinsic coordinates of the complex plane endowed with a conformal metric. This new approach allows us to derive the algebraic equations that characterize relative equilibria. The second part of the paper brings new results about necessary and sufficient conditions for the existence of relative equilibria in the cases n = 2 and n = 3.

/pdf/an-intrinsic-approach-in-the-curved-n-body-problem-the-435j2ooju5.pdf

Ernesto Pérez-Chavela

Papers

The n-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

The n -Body Problem in Spaces of Constant Curvature. Part II: Singularities

THE n-BODY PROBLEM IN SPACES OF CONSTANT CURVATURE

Homographic solutions of the curved 3-body problem

An intrinsic approach in the curved n-body problem. The positive curvature case