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Liouville–Green Method
In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mechanics in which the wave function is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly. The name is an initialism for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used letter combinations include JWKB and WKBJ, where the "J" stands for Jeffreys. Brief history This method is named after physicists Gregor Wentzel, Hendrik Anthony Kramers, and Léon Brillouin, who all developed it in 1926. In 1923, mathematician Harold Jeffreys had developed a general method of approximating solutions to linear, second-order differential equations, a class that includes the Schrödinger equation. The Schrödinger e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms. Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating of the human heart (for circulation), business cycles in economics, predator–prey population cycles in ecology, geothermal geysers in geology, vibration of strings in guitar and other string instruments, periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy. The term ''vibration'' is precisely used to describe a mechanical oscillation. Oscillation, especially rapid oscillation, may be an undesirable phenomenon in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action (physics)
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action close to the Planck constant, quantum effects are significant. In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which it has that amount of energy. More formally, action is a mathematical functional which takes the trajectory ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WKB Approximation To Probability Density
In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mechanics in which the wave function is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly. The name is an initialism for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used letter combinations include JWKB and WKBJ, where the "J" stands for Jeffreys. Brief history This method is named after physicists Gregor Wentzel, Hendrik Anthony Kramers, and Léon Brillouin, who all developed it in 1926. In 1923, mathematician Harold Jeffreys had developed a general method of approximating solutions to linear, second-order differential equations, a class that includes the Schrödinger equation. The Schrödinger e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergent Series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eikonal Equation
An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation. The classical eikonal equation in geometric optics is a differential equation of the form where x lies in an open subset of \mathbb^n, n(x) is a positive function, \nabla denotes the gradient, and , \cdot , is the Euclidean norm. The function n is given and one seeks solutions u . In the context of geometric optics, the function n is the refractive index of the medium. More generally, an eikonal equation is an equation of the form where H is a function of 2n variables. Here the function H is given, and u is the solution. If H(x,y)= , y, - n(x) , then equation () becomes (). Eikonal equations naturally arise in the WKB method and the study of Maxwell's equations. Eikonal equations provide a link between physical (wave) optics and geometric (ray) optics. One fast computational algorithm t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Dominant Balance
In mathematics, the method of dominant balance approximates the solution to an equation by solving a simplified form of the equation containing 2 or more of the equation's terms that most influence (dominate) the solution and excluding terms contributing only small modifications to this approximate solution. Following an initial solution, iteration of the procedure may generate additional terms of an asymptotic expansion providing a more accurate solution. An early example of the dominant balance method is the Newton polygon method. Newton developed this method to find an explicit approximation for an algebraic function. Newton expressed the function as proportional to the independent variable raised to a power, retained only the lowest-degree polynomial terms (dominant terms), and solved this simplified reduced equation to obtain an approximate solution. Dominant balance has a broad range of applications, solving differential equations arising in fluid mechanics, plasma phy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leading-order
The leading-order terms (or leading-order corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.math.ucdavis.edu/~hunter/notes/asy.pdf The sizes of the different terms in the equation(s) will change as the variables change, and hence, which terms are leading-order may also change. A common and powerful way of simplifying and understanding a wide variety of complicated mathematical models is to investigate which terms are the largest (and therefore most important), for particular sizes of the variables and parameters, and analyse the behaviour produced by just these terms (regarding the other smaller terms as negligible). This gives the main behaviour – the true behaviour is only small deviations away from this. The main behaviour may be captured sufficiently well by just the strictly leading-order terms, or it may be decided that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steven Orszag
Steven Alan Orszag (February 27, 1943 – May 1, 2011) was an American mathematician. Life and career Orszag was born to a Jewish family in Manhattan, the son of Joseph Orszag, a lawyer.University of St Andrews, Scotland - School of Mathematics and Statistics: "Steven Alan Orszag" by J.J. O'Connor and E.F. Robertson October 2011 Orszag's paternal grandparents were emigrants from . Orszag was raised in |
Carl M
Carl may refer to: *Carl, Georgia, city in USA *Carl, West Virginia, an unincorporated community *Carl (name), includes info about the name, variations of the name, and a list of people with the name *Carl², a TV series * "Carl", an episode of television series ''Aqua Teen Hunger Force'' * An informal nickname for a student or alum of Carleton College CARL may refer to: *Canadian Association of Research Libraries *Colorado Alliance of Research Libraries See also *Carle (other) *Charles *Carle, a surname *Karl (other) *Karle (other) Karle may refer to: Places * Karle (Svitavy District), a municipality and village in the Czech Republic * Karli, India, a town in Maharashtra, India ** Karla Caves, a complex of Buddhist cave shrines * Karle, Belgaum, a settlement in Belgaum ... {{disambig ja:カール zh:卡尔 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Scale Analysis
In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove (unwanted) secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions. Mathematics research from about the 1980s proposes that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see center manifold and slow manifold). Example: undamped Duffing equation Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
