Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E. The geodesic flow on the full configuration space $C^3$ (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space $C^2$ and shape space $R^3$ (as well as $S^3$ and the shape sphere $S^2$ for the inverse-square potential when E = 0). The corresponding Riemannian submersions are described explicitly in `Hopf' coordinates which are particularly adapted to the isometries. For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy `pair of pants' JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on $C^2, R^3$ & $S^3$ with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits and observe that the geodesic reformulation `regularizes' pairwise and triple collisions on $C^2$ and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and E=0, we find that the scalar curvature on $C^2$ is strictly negative though it could have either sign on $R^3$. However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.Comment: 26 pages, 16 figures. Published version, typos corrected and references update

Design and Analysis of Absorption Refrigeration System Using H2o + [EMIM] [TFA]

With rapid industrialization and constantly increasing energy consumption, human kind is going to face a growing degradation of environment, if the activities continues as usual. This work is focused to design an absorption refrigeration cycle which gives solution to present cooling problems using low grade heat coming out from the industries. More importance has been given to the ionic liquid based working pairs which are eco-friendly in nature. Ionic Liquids (ILs) are polar compounds which are considered as a combination of organic cations and inorganic anions, and are liquid below 100ºC and have potential to replace common VOCs as solvents in chemical processes. ILs have many favorable attributes such as a low vapor pressure, stability over a large liquid working temperature range, and the ability to be designed to dissolve compounds of interest. This research redresses innovative solutions using ILs for absorption refrigeration cycle. Ionic liquids (ILs) are used as absorbents in absorption refrigeration systems, which present the possibility of overcoming some of the safety and environmental concerns of current systems. In general, absorption refrigeration is attractive since electrical energy is replaced with low grade heat energy. Many ILs are completely miscible with water, which leads the focus to investigate ILs and water for this application. Moreover, this research work aims at finding out the value of COP for H2O + [EMIM] [TFA], to design an optimal absorption refrigerator and to study the dependence of COP on various parameters. It also carries out study on the economic feasibility of the absorption refrigeration cycle using H2O + [EMIM] [TFA] working pair to compare with the conventional working pairs and compression refrigeration system. It further addresses the roadblocks in the simulation of the absorption refrigeration cycle using water and [EMIM] [TFA]

Fuzzy Translations of Fuzzy H-ideals in Bck/bciBck/bci-algebras

In this paper, the concepts of fuzzy translation to fuzzy H-ideals in BCK/BCI-algebras are introduced. The notion of fuzzy extensions and fuzzy mul-tiplications of fuzzy H-ideals with several related properties are investigated. Also,the relationships between fuzzy translations, fuzzy extensions and fuzzy multiplica-tions of fuzzy H-ideals are investigated.DOI : http://dx.doi.org/10.22342/jims.21.1.200.45-5