History of variational theory

The idea that laws of nature should satisfy a principle of simplicity goes back at least to the Greek philosophers [436], The anthropomorphic concept that the engineering skill of a supreme creator should result in rules of least effort or of most efficient use of resources leads directly to principles characterized by mathematical extrema. For example, Aristotle (De Caelo) concluded that planetary orbits must be perfect circles, because geometrical perfection is embodied in these curves ... of lines that return upon themselves the line which bounds the circle is the shortest. That movement is swiftest which follows the shortest line . Hero of Alexandria ( Catoptrics) proved perhaps the first scientific minimum principle, showing that the path of a reflected ray of light is shortest if the angles of incidence and reflection are equal. 

The superiority of circular planetary orbits became almost a religious dogma in the Christian era, intimately tied to the idea of the perfection of God and of His creations. It was replaced by modem celestial mechanics only after centuries in which the concept of esthetic perfection of the universe was gradually superseded by a concept of esthetic perfection of a mathematical theory that could account for the 

Variational principles have turned out to be of great practical use in modem theory. They often provide a compact and general statement of theory, invariant or covariant under transformations of coordinates or functions, and can be used to formulate internally consistent computational algorithms. Symmetry properties are often most easily derived in a variational formalism. 

The law of geometrical optics anticipated by Hero of Alexandria was formulated by Fermat (1601-1655) as a principle of least time, consistent with Snell s law of refraction (1621). The time for phase transmission from point P to point Q along a path x(r) is given by 

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This part is concerned with variational theory prior to modem quantum mechanics. The exception, saved for Chapter 10, is electromagnetic theory as formulated by Maxwell, which was relativistic before Einstein, and remains as fundamental as it was a century ago, the first example of a Lorentz and gauge covariant field theory. Chapter lisa brief survey of the history of variational principles, from Greek philosophers and a religious faith in God as the perfect engineer to a set of mathematical principles that could solve practical problems of optimization and rationalize the laws of dynamics. Chapter 2 traces these ideas in classical mechanics, while Chapter 3 discusses selected topics in applied mathematics concerned with optimization and stationary principles. 

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

Much of the variation in these time series for the past 700 kyr can be described by a combination of a 100 kyr cycle plus additional cycles with periods of 20 and 40 kyr. This result immediately suggests that the ice-age cycles are caused by variations in the amount and seasonality of solar radiation reaching the Earth (insolation), because the 20, 40, and 100 kyr periods of climate history match the periods of cyclic variations in Earth s orbit and axial tilt, line hypothesis that these factors control climate was proposed by Milutin Milankovitch in the early part of the 20th century and is widely known as "Milankovitch Theory." It is now generally accepted that the Milankovitch variations are the root cause of the important 20 and 40 kyr climate cycles. The 100 kyr cycle, however, proves to be a puzzle. The magnitude of the insolation variation at this periodicity is relatively trivial, but the 100 kyr cycle dominates the climate history of the last 700 kyr. Further,... 

Population genetics has produced a detailed theory of the relationship between frequency, drift, and selective value of genetic variation (Nei, 1975 Hard and Clark, 1989 Li, 1997). The problem with its straightforward application is that the main parameters, such as history of population size, fixation rate, and selective pressure, are difficult to establish. 

I consider three sources of variation in behavior across societies. First, societies may exhibit different behaviors because for historical reasons they have ended up in different coordination equilibria. In theory, the United Kingdom and the United States could be identical in all respects except that people drive on the left in the first country and on the right in the second, together with features that flow directly from this difference. Differences in metric systems, number systems, and calendars can also generate differences in behavior that need not reflect anything but accidents of history. Second, behavioral differences may be due to different norms and values. I especially emphasize the importance of social norms in explaining cultural variation. Norms, in turn, are backed by... 

In Darwin s times heredity was a mystery, but this did not prevent him from concluding that natural selection works on heritable variations. All that he needed to know about heredity were the two facts that he learned from breeders, namely that (1) every individual in a population has unique characteristics, and (2) many distinctive traits are inherited. The discovery of the hereditary mechanism could not cancel these experimental facts, and could not therefore deny natural selection. That discovery, however, could reveal new mechanisms of evolution, and reduce the role that natural selection played in the history of life. This is why the study of heredity came to be seen as the testing ground for any evolutionary theory, and for almost a century, in fact, the debate on evolution has largely been a debate on genetics. 

During the geological formation of major lead deposits the lead is separated from the parent uranium and thorium, and the isotopic composition of the lead so separated depends on the geochemical history of the source materials, specifically on the U/Pb and Th/Pb ratios and the timing of any variations in these ratios including the time of the final separation of lead. In theory, it is possible for each lead deposit in the world to have its own unique isotopic composition. 

Wiley, Chichester (Novartis Found Symp 223) p 260-275 Rice WR1988 Heritable variation in fitness as a prerequisite for adaptive female choice the effect of mutation-selection balance. Evolution 42 817-820 Rose M, Charlesworth B 1980 A test of evolutionary theories of senescence. Nature 287 141—142 Rushton JP 1995 Race, evolution, and behavior a life history perspective. Transaction, New Brunswick, NJ... 

If, at the temperature corresponding with the liquid be rapidly cooled, separation of solid solution, will take place at the temperature corresponding with 4 That is, the freezing-point varies with the rapidity of cooling. Further, the solid solution is an equilibrium mixture of Pa and Pft and this equilibrium may alter with the temperature, as shown by the line Sih, Consequently, if white phosphorus is heated rapidly from a low temperature after the internal equilibrium has been established, the melting-point will not be but, say, % The diagram enables one, therefore, on the basis of the theory put forward by Smits, to account for the variation of physical properties, e,g, melting-point, with the past history and treatment of the substance. 

As varied as the interests of landmark entrepreneur and scientist Alfred Nobel, the present book Quantum Atom and Periodicity by Mihai V. Putz, explores the story of periodieity of ehemieal elements by ineorporating topies spanning from history to topies as eomplex as Feynman-Kleinert variational theory, density fimetional theory as well as quantum theory of periodicity. The author has very lueely established a coimeetion between the elassieal views and quantum theory about periodieity of elements. With its clear descriptions and explanations, readers from both physics as well as chemistry can appreciate and comprehend the book. 

Figure 8. The fragility parameter m is plotted as a function of the NMT nonlinearity parameter Xnmt- The curve is predicted by the RFOT theory when the temperature variation of >o is neglected. The data are taken from Ref. [49]. The disagreement may reflect a breakdown of phenomenology for the history dependence of sample preparation. The more fragile substances consistently lie above the prediction, which has no adjustable parameters. This discrepancy may be due to softening effects.

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