Search Results (5)

We study a class of circular restricted planar Newtonian four-body problems in which three masses are positioned at the vertices of a Lagrange equilateral triangle configuration, each mass revolving around the center of mass in circular orbits. Assuming that the value of the fourth mass is negligibly small (i.e., it does not perturb the motion of the other three masses, though its own motion is influenced by them), we use variational minimization methods to prove the existence of noncollision periodic solutions with some fixed winding numbers. These noncollision solutions exist for both equal and unequal mass values for the three bodies located at the vertices of the Lagrange equilateral configuration. Full article

The graph theory-based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the “momenta graph”. According to the Ramsey theorem, this graph contains at least one mono-chromatic triangle. This is true even for chaotic motion of three bodies; thus, illustrating the idea supplied by the Ramsey theory, total chaos is impossible. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereographic momenta graph is introduced. Shannon entropy of the momenta graph is calculated. The particular configurations of bodies are addressed, including the Lagrange configuration and the figure eight-shaped motion. The suggested approach is generalized for the quantum field theory with the Pauli–Lubanski pseudo-vector. The suggested coloring procedure is the Lorenz invariant. Full article

To observe lower-frequency gravitational waves (GWs), it is effective to utilize a large spacecraft formation baseline, spanning hundreds of thousands to millions of kilometers. To overcome the limitations of a gravitational-wave observatory (GWO) on specific orbits, a scientific observation mode and a non-scientific observation mode for GWOs are proposed. For the non-scientific observation mode, this paper designs equilateral triangle and equilateral tetrahedral array formations for a space-based GWO near a collinear libration point. A stable configuration is the prerequisite for a GWO; however, the motion near the collinear libration points is highly unstable. Therefore, the output regulation theory is applied. By leveraging the tracking aspect of the theory, the equilateral triangle and equilateral tetrahedral array formations are achieved. For an equilateral triangle array formation, two geometric configuration design methods are proposed, addressing the fuel consumption required for initialization and maintenance. To observe GWs in different directions and avoid configuration/reconfiguration, the multi-layer equilateral tetrahedral array formation is given. Additionally, the control errors are calculated. Finally, the effectiveness of the control method is demonstrated using the Sun–Earth circular-restricted three-body problem (CRTBP) and the ephemeris model located at Lagrange point $L_1$. Full article

The time evolution of the continuous measure of symmetry for a system built of three bodies interacting via the potential $U\left(r\right)~\frac{1}{r}$ is reported. Gravitational and electrostatic interactions between the point bodies were addressed. In the case of a pure gravitational interaction, the three-body-system deviated from its initial symmetrical location, described by the Lagrange equilateral triangle, comes eventually to collapse, accompanied by the growth of the continuous measure of symmetry. When three point bodies interact via the repulsive Coulomb interaction, the time evolution of the CMS is quite different. The CMS calculated for all of the studied initial configurations of the point charges, and all of their charge-to-mass ratios, always comes to its asymptotic value with time, evidencing the stabilization of the shape of the triangle, constituted by the interacting bodies. The influence of Stokes-like friction on the change in symmetry of three-body gravitating systems is elucidated; the Stokes-like friction slows the decrease in the CMS and increases the stability of the Lagrange triangle. Full article

With the integrated development of the Internet, wireless sensor technology, cloud computing, and mobile Internet, there has been a lot of attention given to research about and applications of the Internet of Things. A Wireless Sensor Network (WSN) is one of the important information technologies in the Internet of Things; it integrates multi-technology to detect and gather information in a network environment by mutual cooperation, using a variety of methods to process and analyze data, implement awareness, and perform tests. This paper mainly researches the localization algorithm of sensor nodes in a wireless sensor network. Firstly, a multi-granularity region partition is proposed to divide the location region. In the range-based method, the RSSI (Received Signal Strength indicator, RSSI) is used to estimate distance. The optimal RSSI value is computed by the Gaussian fitting method. Furthermore, a Voronoi diagram is characterized by the use of dividing region. Rach anchor node is regarded as the center of each region; the whole position region is divided into several regions and the sub-region of neighboring nodes is combined into triangles while the unknown node is locked in the ultimate area. Secondly, the multi-granularity regional division and Lagrange multiplier method are used to calculate the final coordinates. Because nodes are influenced by many factors in the practical application, two kinds of positioning methods are designed. When the unknown node is inside positioning unit, we use the method of vector similarity. Moreover, we use the centroid algorithm to calculate the ultimate coordinates of unknown node. When the unknown node is outside positioning unit, we establish a Lagrange equation containing the constraint condition to calculate the first coordinates. Furthermore, we use the Taylor expansion formula to correct the coordinates of the unknown node. In addition, this localization method has been validated by establishing the real environment. Full article