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Central Force Optimization
Published in Nazmul Siddique, Hojjat Adeli, Nature-Inspired Computing, 2017
According to classical mechanics, a central force acting on an object is defined by the force that points radially, the magnitude of which depends only on the distance of the object from the origin and is directed along the line joining them (Whittaker, 1944). The central force problem is an important problem in classical physics. Many naturally occurring forces are central (Goldstein, 1980) such as gravitational force described by Newton's law of gravity and electromagnetic force described by Coulomb's law. The concept of CFO is based on gravitational force where masses move under the influence of gravity. In CFO, a set of “probes” fly through a decision space (DS) according to two simple equations derived from the gravitational metaphor. Probes are analogous to agents or particles used in many other meta-heuristic algorithms. CFO is inherently deterministic, unlike other widely used meta-heuristics.
Central force problems
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
In this chapter we will be interested in the problem of the central force in which the force function depends only on the distance r from the origin thus embodying the character of spherical symmetry. In some of the well-known examples of the central force like Newton’s inverse square law of gravitation and Coulomb’s electrostatic force between two charges, the force function F (r) is proportional to the inverse square of the distance and is negative. However, in the case of the spherical harmonic oscillator, F (r) is linearly proportional to the distance and negative as well. By Bertrand’s theorem1 it can be established that these two force functions are the only possible forms of the central force fields having orbits that are all stable and closed. In the central force problem the underlying force acts in a direction that is toward or away from a fixed point called the “force center.” As such, the torque r→×F→ on the particle about the force center vanishes resulting in the constancy of the angular momentum Ω→=r→×(mv→)=r→×p→.
Why Are Rockets Needed?
Published in Travis S. Taylor, Introduction to Rocket Science and Engineering, 2017
Figure 2.14 shows a spacecraft in an elliptical orbit about the Earth. The force due to gravity acting on the spacecraft is radially inward toward the center of the Earth. Any force that acts toward or away from a fixed point radially is called a central force. The equation for describing the torque on the spacecraft due to a central force is
A new interpretation of the experimental data for the OH+SO collision considering the recrossing reaction
Published in Molecular Physics, 2020
Juan de Dios Garrido, Samah Ellakkis, Maikel Y. Ballester
The most simple approach of the mentioned theories assumes that interaction between the elements in collision is produced by a central force with an associated central potential energy [2,4,13,14], therefore, according to classical mechanics [15], the phenomenon can be studied as a one-dimensional problem. Following this idea, the collision between an atom and a diatomic molecule, or two diatomic molecules, interacting with a central force (of the type of inverse-power law), is simplified to the motion of a representative particle with reduced mass (here and are the masses of colliding particles), under the action of the effective central potential energy [4] In this equation, r is the distance between the positions of the representative particle and the force centre (that is taken as the origin of coordinates), s is a value that depends upon the interaction type, is a constant, also related to the type of interaction, is the initial translational energy (named as collision energy [2,4]) of the representative particle and b is the impact parameter. The first term in (1) is the well known centrifugal barrier, with a repulsive character, and the second one is the general attractive potential energy.
Perspectives on geometric numerical integration
Published in Journal of the Royal Society of New Zealand, 2019
A diagram from Book I, Theorem I of Newton's Principia is shown in Figure 1. It proves that a body moving under a central force sweeps out equal areas in equal times, or, in modern terms, the conservation of angular momentum for motion in a central force field. Since 1918 this property has been understood as an example of Noether's theorem, a cornerstone result in geometric mechanics. Newton has discretised the smooth motion of the planet into a repeated sequence of two steps (‘For suppose the time to be divided into equal parts’.). In the first step (AB), the planet moves in a straight line. In the second step, the planet does not move, but its velocity vector is changed under the influence of the central force from Bc to BC. The triangles SAB and SBC have the same area. He then lets the step size tend to zero (‘Now let the number of those triangles be augmented … ad infinitum’), arriving at the desired result for the continuous system.
Electronic structure, mechanical and optical properties of ternary semiconductors Si1-xGexC (X = 0, 0.25, 0.50, 0.75, 1)
Published in Philosophical Magazine, 2019
M. Manikandan, A. Amudhavalli, R. Rajeswarapalanichamy, K. Iyakutti
The mechanical stability of Si1-xGexC (X = 0, 0.25, 0.50, 0.75, 1) in cubic zinc blende (β) phase and (X = 0, 0.33, 0.50, 0.67, 1) in hexagonal (α) phase is analysed. The complete set of zero pressure elastic constants for these structures calculated by strain–stress method [33,34] are shown in Tables 3 and 4. The elastic constants of Si1-xGexC in cubic and hexagonal phases satisfied the Born-Huang elastic stability criteria [35] and hence these ternary carbides are mechanically stable at ambient condition. Using Voigt–Reuss–Hill approximation [29–31], the bulk modulus (B) and shear modulus (G) are calculated. The calculated bulk modulus values are comparable with the available data [1–9]. Among these semiconducting carbides, SiC has the highest bulk modulus (215 GPa) and Si0.50Ge0.50C has the largest shear modulus value (174 GPa) in cubic (β) phase. In hexagonal phase, SiC possesses the highest bulk modulus value 327 GPa and shear modulus value of 333 GPa. The lower and upper limits of Poisson’s ratio for central force field in solids are 0.25 and 0.5 [36]. The obtained Poisson’s ratio for zinc blende (cubic) GeC is close to the value of 0.25, which indicates that GeC has central inter atomic forces.