$4+1$-Moulton Configuration and Positive Mass Deformation

2021 ◽  
pp. 286-300
Author(s):  
Naoko Yoshimi ◽  
Akira Yoshioka
Keyword(s):  
Inverse Problem ◽  
Explicit Formula ◽  
Body Problem ◽  
Central Configuration ◽  
Positive Mass ◽  
Mass Deformation

For given $k$ bodies of collinear central configuration of Newtonian $k$-body problem, we ask whether one can add another body on the line without changing the configuration and motion of the initial bodies so that the total $k+1$ bodies provide a central configuration. The case $k=4$ is analyzed. We study the inverse problem of five bodies and obtain a global explicit formula. Then using the formula we find there are five possible positions of the added body and for each case the mass of the added body is zero. We further consider to deform the position of the added body without changing the positions of the initial four bodies so that the total five bodies are in a state of central configuration and the mass of the added body becomes positive. For each solution above, we find such a deformation of the position of the added body in an explicit manner starting from the solution.

scholarly journals Regions of Central Configurations in a Symmetric 4 + 1-Body Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Shoaib
Keyword(s):  
Inverse Problem ◽  
Body Problem ◽  
Center Of Mass ◽  
Central Configurations ◽  
Central Configuration ◽  
Isosceles Trapezoid ◽  
The Masses

The inverse problem of central configuration of the trapezoidal 5-body problems is investigated. In this 5-body setup, one of the masses is chosen to be stationary at the center of mass of the system and four-point masses are placed on the vertices of an isosceles trapezoid with two equal massesm1=m4at positions∓0.5, rBandm2=m3at positions∓α/2,rA. The regions of central configurations where it is possible to choose positive masses are derived both analytically and numerically. It is also shown that in the complement of these regions no central configurations are possible.

scholarly journals Central configurations in planar n-body problem with equal masses for $$n=5,6,7$$

2019 ◽  
Vol 131 (10) ◽  
Author(s):  
Małgorzata Moczurad ◽  
Piotr Zgliczyński
Keyword(s):  
Body Problem ◽  
Central Configurations ◽  
Central Configuration ◽  
Newtonian Potential ◽  
Computer Assisted ◽  
Computer Assisted Proof ◽  
N Body Problem ◽  
Reflective Symmetry

Abstract We give a computer-assisted proof of the full listing of central configuration for n-body problem for Newtonian potential on the plane for $$n=5,6,7$$ n = 5 , 6 , 7 with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For $$n=8,9,10$$ n = 8 , 9 , 10 , we establish the existence of central configurations without any reflectional symmetry.

CLASSIFICATION OF OBJECTS IN AN ACOUSTIC WAVEGUIDE BY INVERSION OF THE FARFIELD DATA

2006 ◽  
Vol 14 (02) ◽  
pp. 185-199 ◽  
Author(s):  
DOO-SUNG LEE ◽  
R. P. GILBERT ◽  
NOAM ZEEV
Keyword(s):  
Inverse Problem ◽  
Analytical Solution ◽  
Least Squares ◽  
Scattered Field ◽  
Body Problem ◽  
Acoustic Fields ◽  
Parallel Plates ◽  
Tangent Sphere ◽  
Unknown Object

In this paper we investigate the unknown body problem in a waveguide. The Rayleigh conjecture states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. This conjecture is the cornerstone of the intersecting canonical body approximation ICBA for solving the unknown body inverse problem. Therefore, the use of the ICBA requires that an analytical solution be known exterior to the sphere in the waveguide, which leads us to analytically compute the exterior solution for a sphere between two parallel plates. A least-squares matching of theoretical acoustic fields against the measured, scattered field permits a reconstruction of the unknown object.

scholarly journals Symmetry of planar four-body convex central configurations

2008 ◽  
Vol 464 (2093) ◽  
pp. 1355-1365 ◽  
Author(s):  
Alain Albouy ◽  
Yanning Fu ◽  
Shanzhong Sun
Keyword(s):  
Body Problem ◽  
The Other ◽  
Central Configurations ◽  
Central Configuration ◽  
Geometric Properties ◽  
Four Body Problem ◽  
Planar Central Configurations ◽  
The Masses ◽  
The Relationship ◽  
Convex Central Configuration

We study the relationship between the masses and the geometric properties of central configurations. We prove that, in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.

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