China
Africa
Explore chapters and articles related to this topic
Introduction
Published in Desineni Subbaram Naidu, Optimal Control Systems, 2018
The calculus of variations is that branch of mathematics that deals with finding a function which is an extremum (maximum or minimum) of a functional. A functional is loosely defined as a function of a function. The theory of finding maxima and minima of functions is quite old and can be traced back to the isoperimetric problems considered by Greek mathematicians such as Zenodorus (495–435 B.C.) and by Poppus (c. 300 A.D.). But we will start with the works of Bernoulli. In 1699 Johannes Bernoulli (1667–1748) posed the brachistochrone problem: the problem of finding the path of quickest descent between two points not in the same horizontal or vertical line. This problem which was first posed by Galileo (1564–1642) in 1638, was solved by John, his brother Jacob (1654–1705), by Gottfried Leibniz (1646–1716), and anonymously by Isaac Newton (1642–1727). Leonard Euler (1707–1783) joined John Bernoulli and made some remarkable contributions, which influenc d Joseph-Louis Lagrange (1736–1813), who finally gave an elegant way of solving these types of problems by using the method of (first) variations. This led Euler to coin the phrase calculus of variations. Later this necessary condition for extrema of a functional was called the Euler - the Lagrange equation. Lagrange went on to treat variable end - point problems introducing the multiplier method, which later became one of the most powerful tools-Lagrange (or Euler-Lagrange) multiplier method-in optimization.
Laplace Transforms
Published in Steven G. Krantz, Differential Equations, 2015
Thus the tautochrone turns out to be a cycloid. This problem and its solution is one of the great triumphs of modern mechanics. An additional very interesting property of this curve is that it is the brachistochrone. That means that, given two points A and B in space, the curve connecting them down which a bead will slide the fastest is the cycloid (Figure 8.4). This last assertion was proved by Isaac Newton, who read the problem as posed in a public challenge by Bernoulli in a periodical. Newton had just come home from a long day at the British Mint (where he worked after he gave up his scientific work). He solved the problem in a few hours, and submitted his solution anonymously. But Bernoulli said he knew it was Newton; he “recognized the lion by his claw.”
Extrema and Variational Calculus
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
By the seventeenth century, mathematicians were interested in paths of quickest descent. Galileo noted, “From the preceding it is possible to infer that the path of quickest descent from one point to another is not the shortest path, namely, a straight line, but the arc of a circle.” The curves that interested people of that time were the cycloid, the tautochrone (or, isochrone), and the brachistochrone. These are simply defined in terms of simple motions along curves. The cycloid is the curve traced by a point on the circumference of a wheel that rolls without slipping.The brachistochrone is the curve along which a free-sliding particle will descend more quickly between two given points than on any other AB-curve.The tautochrone is the curve along which a particle has a descent time independent of the initial position.
Design, Fabrication & Analysis of a Gravitational Water Vortex Based Energy Harvester
Published in International Journal of Green Energy, 2023
Zulfikre Esa, Juliana Hj Zaini, Murtuza Mehdi, Asif Iqbal, Malik Muhammad Nauman
Within the context of this research study, the design parameters are mainly focused on the dome basin shapes and its parameters. As mentioned above, the model design was inspired by the Brachistochrone curve theory that derives the fastest path between two elevated points, i.e. curved path and straight line path (Nishiyama 2013). CFD analysis has been done as a preliminary study for developing and predicting the dome basin model and its performance. It was used to determine the optimized configuration of the parameters by understanding and examining the flow pattern with the developed 3D model of GWVPP system. The optimized parameters achieved from the CFD analysis were selected for fabrication. The dome-basin configuration is modeled with assumption that the air-core vortex is constant and axisymmetric while the flow is incompressible in cylindrical coordinates (Dhakal et al. 2015b, Thapa, Mishra, and Sarath 2017, Dhakal et al. 2014). The CAD drawing of the dome-basin model and the parameters of interest are shown in Figure 1. The selected parameters for designing the GWVPP system in this paper were based on a combination of previous researchers’ optimized parameter settings as presented in Table 1 (Mulligan and Hull 2010, Raveendran et al. 2016, Power, McNabola, and Coughlan 2015, Dhakal et al. 2014).
The brachistochrone problem revisited: a numerical solution via non-linear root-finding
Published in International Journal of Mathematical Education in Science and Technology, 2022
Jonathan Hoseana, Maria Regina Kusnadi, Ivander Jeremy, Felisha Felisha
In May 1697, the same journal published solutions by Leibniz, Newton, Jakob Bernoulli (a brother of Johann Bernoulli) and Johann Bernoulli himself (Babb & Currie, 2008; Freguglia & Giaquinta, 2016), which simultaneously reveal that the desired path takes the form of a cycloid. The brachistochrone problem is the term popularly used to refer to this problem (Babb & Currie, 2008; Coleman, 2012; Erlichson, 1999; Freguglia & Giaquinta, 2016; Sahab et al., 2013), etymologically originating from the Greek words brachistos, meaning the shortest, and chronos, meaning time (Sahab et al., 2013, p. 26). The solution to this problem has practical implications for designers of, e.g. slides and rollercoaster tracks (Babb & Currie, 2008, p. 172).