scholarly journals Central Configurations for NewtonianN+2p+1-Body Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Furong Zhao ◽  
Jian Chen
Keyword(s):  
Central Configurations ◽  
Straight Line ◽  
Spatial Central Configurations ◽  
The Masses

We show the existence of spatial central configurations for theN+2p+1-body problems. In theN+2p+1-body problems,Nbodies are at the vertices of a regularN-gonT;2pbodies are symmetric with respect to the center ofT, and located on the straight line which is perpendicular to the regularN-gonTand passes through the center ofT; theN+2p+1th is located at the the center ofT. The masses located on the vertices of the regularN-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center ofTare equal.

scholarly journals Spatial Central Configurations with Two Twisted Regular 4-Gons

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Sen Zhang ◽  
Furong Zhao
Keyword(s):  
Twist Angle ◽  
Central Configurations ◽  
Central Configuration ◽  
Spatial Central Configurations ◽  
The Masses ◽  
Axis Passing

We study the configuration formed by two squares in two parallel layers separated by a distance. We picture the two layers horizontally with thez-axis passing through the centers of the two squares. The masses located on the vertices of each square are equal, but we do not assume that the masses of the top square are equal to the masses of the bottom square. We prove that the above configuration of two squares forms a central configuration if and only if the twist angle is equal tokπ/2or (π/4+kπ/2)(k=1,2,3,4).

scholarly journals On the Existence of Central Configurations of2k+2p+2l-Body Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yueyong Jiang ◽  
Furong Zhao
Keyword(s):  
Central Configurations ◽  
The Masses

We prove the existence of central configurations of the2k+2p+2l-body problems with Newtonian potentials inR3. In such configuration,2kmasses are symmetrically located on thez-axis,2pmasses are symmetrically located on they-axis, and2lmasses are symmetrically located on thex-axis, respectively; the masses symmetrically about the origin are equal.

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.

Dynamics of a Self-Propelling Wheel With Three Eccentric Masses

2000 ◽  
Author(s):  
Tuhin K. Das ◽  
Ranjan Mukherjee
Keyword(s):  
Potential Energy ◽  
Constant Velocity ◽  
Simple Approach ◽  
Constant Acceleration ◽  
Physical Constraints ◽  
Straight Line ◽  
Radial Spokes ◽  
Simulation Results ◽  
The Masses ◽  
Variable Potential

Abstract This paper investigates the dynamics of a rolling disk with three unbalance masses that can slide along radial spokes equispaced in angular orientation. The objective is to design trajectories for the masses that satisfy physical constraints and enable the disk to accelerate or move with constant velocity. The disk is designed to remain vertically upright and is constrained to move along a straight line. We design trajectories for constant acceleration through detailed analysis using a dynamic model. The analysis considers two separate cases; one where the potential energy of the system is conserved, and the other where it continually varies. Whereas trajectories conserving potential energy are limacons, the variable potential energy trajectories are the most general and allow greater acceleration. Following the strategy for constant acceleration maneuvers, we give a simple approach to tracking an acceleration profile and provide simulation results.

scholarly journals New Classes of Spatial Central Configurations forN+N+ 2-Body Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Liu Xuefei ◽  
Zhang Chuntao ◽  
Luo Jianmei ◽  
Zhang Gan
Keyword(s):  
Body Problem ◽  
Central Configurations ◽  
Spatial Central Configurations

scholarly journals New classes of spatial central configurations for the 7-body problem

2014 ◽  
Vol 86 (1) ◽  
pp. 3-9
Author(s):  
ANTONIO CARLOS FERNANDES ◽  
LUIS FERNANDO MELLO
Keyword(s):  
Body Problem ◽  
Central Configurations ◽  
Equilateral Triangles ◽  
Spatial Central Configurations

In this paper we show the existence of new families of spatial central configurations for the 7-body problem. In the studied spatial central configurations, six bodies are at the vertices of two equilateral triangles , and one body is located out of the parallel distinct planes containing and . The results have simple and analytic proofs.

scholarly journals Extended Lagrangian formalism for rheonomic systems with variable mass

2017 ◽  
Vol 44 (1) ◽  
pp. 115-132 ◽  
Author(s):  
Djordje Musicki
Keyword(s):  
Energy Conservation ◽  
Variable Mass ◽  
Frame Of Reference ◽  
Lagrangian Formalism ◽  
Extended System ◽  
Generalized Coordinates ◽  
Lagrangian Equations ◽  
Straight Line ◽  
The Masses ◽  
Energy Relations

In this paper the extended Lagrangian formalism for the rheonomic systems (Dj. Musicki, 2004), which began with the modification of the mechanics of such systems (V. Vujicic, 1987), is extended to the systems with variable mass, with emphasis on the corresponding energy relations. This extended Lagrangian formalism is based on the extension of the set of chosen generalized coordinates by new quantities, suggested by the form of nonstationary constraints, which determine the position of the frame of reference in respect to which these generalized coordinates refer. As a consequence, an extended system of the Lagrangian equations is formulated, accommodated to the variability of the masses of particles, where the additional ones correspond to the additional generalized coordinates. By means of these equations, the energy relations of such systems have been studied, where it is demonstrated that here there are four types of energy conservation laws. The obtained energy laws are more complete and natural than the corresponding ones in the usual Lagrangian formulation for such systems. It is demonstrated that the obtained energy laws, are in full accordance with the energy laws in the corresponding vector formulation, if they are expressed in terms of the quantities introduced in this formulation of mechanics. The obtained results are illustrated by an example: the motion of a rocket, which ejects the gasses backwards, while this rocket moves up a straight line on an oblique plane, which glides uniformly in a horizontal direction.

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