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Topic: Variational principle

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	WWU Math Department - Colloquium
		Abstract: As is well known, a variational description of a system is very desirable both from mathematical and from physical points of view.
		In the Herglotz variational principle the functional, whose extrema are sought, is defined by a differential equation instead of the classical variational integral.
		This variational principle is important for a number of reasons.
	www.wwu.edu /depts/math/colloquium/c_012705.htm (199 words)

	Variational principle - Wikipedia, the free encyclopedia
		A variational principle is a principle in physics which is expressed in terms of the calculus of variations.
		In what is referred to in physics as Noether's theorem, the Poincaré group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.
		The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.
	en.wikipedia.org /wiki/Variational_principle (400 words)

	Variational principle
		A variational principle is a principle in physics which is expressed in terms of the calculus of variations.
		The principle of least action in mechanics, electromagnetic theory, and quantum mechanics.
		The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.
	www.brainyencyclopedia.com /encyclopedia/v/va/variational_principle.html (619 words)

	Calculus Of Variations (Site not responding. Last check: 2007-11-04)
		Calculus of variations is a field of mathematics which deals with functionss of functions, as opposed to ordinary calculus which deals with functions of numbers.
		Variational methods are important in theoretical physics: in Lagrangian mechanics and in application of the principle of stationary action to quantum mechanics.
		The study of geodesics in differential geometry is a field with an obvious variational content.
	www.wikiverse.org /calculus-of-variations (365 words)

	's Storefront - Lulu.com
		The variational principle of extremum is stated and proved for electromechanical systems of arbitrary configuration wherein the electromagnetic, mechanical, thermal hydraulic and other processes are going on.
		The principle is generalized for the systems described by partial differential equations, and in particular by Maxwell equations.
		The variational optimum principle for electric power systems, which are in essence non-linear electric circuits, is formulated and proven.
	www.lulu.com /solik (564 words)

	Variational principle -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-04)
		A (Click link for more info and facts about variational principle) variational principle is a principle in (The science of matter and energy and their interactions) physics which
		is expressed in terms of the (The calculus of maxima and minima of definite integrals) calculus of variations.
		According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is (Click link for more info and facts about self-adjoint) self-adjoint.
	www.absoluteastronomy.com /encyclopedia/v/va/variational_principle.htm (366 words)

	Variational Principle
		Calculus of variation utilize the idea that variations of a function/functional at its local extrema is zero.
		If x varies with another variable t, one could find all values of t such that the function dependent on x which therefore depends on t is maximized.
		Calculus of variation is a useful technique used both in classical mechanics and quantum mechanics.
	web.mit.edu /chungc/interest/vp/vp.html (602 words)

	Theoretical methods
		The variational principle is at the heart of most of the methods used.
		The variational principle is very powerful and it is capable of explaining the whole of physics.For instance, you can use it to explain all the phenomena described by classical mechanics without recur to Newton's laws.
		So, if the variational principle can help us to determine the ground state wavefunctions, by minimizing the energy, how is it possible that it also give wave functions for excited states, which certainly don't correspond to the minimum.
	www.hpc.susx.ac.uk /~ricardoe/theoreticalmethods.htm (1017 words)

	THE HAMILTON AND TYPE GENERALIZED VARIATIONAL PRINCIPLE FOR THE COMPOSITE LAMINATED PLATES AND SHELLS (Site not responding. Last check: 2007-11-04)
		Abstract By introducing the Hamilton theory and algorithms into the problems of laminated composite plates and shells, the Hamiltion type generalized variational principle is established, and the Hamilton canonical equations and the boundary conditions for the static and elastoplastic analysis of composite plates are presented.
		With the transformation of phase variables, the Hamilton canonical equations and their boundary conditions for the cylindrical shells and doubly curved shells in the curvilinear coordinate are given.
		By transforming the state-vector variables, the Hamilton canonical equations and boundary conditons for the composite laminated cylindrical shells and doubly curved shells in the curvilinear coordinate will be presented in the paper.
	www.shu.edu.cn /journal/vol1no2199708.htm (901 words)

	Fluid-Structure Interaction (Site not responding. Last check: 2007-11-04)
		Since the variational principles are employed to derive numerical solutions, many researchers have attempted to derive variational principles for different classes of the fluid-structure interaction problems.
		Luke has incorporated a variable boundary in his variational principle in order to generate the governing equations for the special case of gravity waves in an incompressible fluid.
		123] introduced a variational principle, utilizing the stream function, in order to model incompressible flow with a free surface under gravity using the finite element method but did not extend their method to the general compressible fluid-structure problem.
	faculty.evansville.edu /az12/PhdThesis/node22.html (848 words)

	PlanetPhysics: variational principle
		In classical mechanics, the term “variational principle” is used to describe a texhnique wheteby the dynamics of a system may be deduced by extremization of a suitable function.
		To illustrate this notion of variational principle, we may consider the example of a particle on a line moving under the influence of a force derived from a potential
		This is version 5 of variational principle, born on 2005-08-14, modified 2005-10-20.
	planetphysics.org /encyclopedia/Action.html (546 words)

	6.1 Kohn's method
		The discontinuity in the occupation number derivative of the penalty functional at idempotency is required because of the non-variational behaviour of the total energy with respect to these variations (section 4.2).
		The behaviour of the penalty functional for unconstrained occupation number variation is plotted in figure 6.1, and in figure 6.2 the total functional is sketched schematically for several representative values of the parameter
		The variational property of the total functional is that it is minimal at the ground-state, but this minimum is defined in terms of the functional taking its minimum value there, not in terms of a vanishing gradient (the gradient being undefined at the ground-state).
	www.tcm.phy.cam.ac.uk /~pdh1001/thesis/node36.html (799 words)

	Variational principles for critical parameters and ionization energies of quantum systems (Site not responding. Last check: 2007-11-04)
		This variational principle is optimized in order to give accurate estimation of the critical parameter itself rather than the energy.
		For ionization energies, an ordinary variational principle for the energy functional was used, but with allowance of complex variational parameters.
		It means that the variational principle produces complex energy that approximates position (real part) and half width (imaginary part) of the corresponding quasi-stationary state.
	www.asergeev.com /files/confs/midwest/abstract.htm (374 words)

	Summary of the method (Site not responding. Last check: 2007-11-04)
		The basic idea of variational analysis is to determine a continuous field approximating the data and exhibiting small spatial variations.
		The first contribution in the variational principle is a measure of the smoothness of the target field.
		In principle, the value of the weight could be adjusted to every observation individually, but in practice it is impossible to decide whether one observation is more reliable than another.
	modb.oce.ulg.ac.be /atlas/node35.html (323 words)

	Variational Principle (Site not responding. Last check: 2007-11-04)
		Yang, Qiang (2004-05-17) Thermomechanical variational principles for dissipative...
		Variational principles for critical parameters and ionization energies of quantu...
		Kohn variational principle for a general finite-range scattering functional...
	www.scienceoxygen.com /phys/179.html (185 words)

	3.3 Variational Monte Carlo
		The variational principle of quantum mechanics, derived in the following section, states that the energy of a trial wavefunction will be greater than or equal to the energy of the exact wavefunction.
		It occurs in both the variational and diffusion Monte Carlo algorithms and its properties are exploited to optimise trial wavefunctions.
		The spatially averaged variance of the local energy is therefore a quantity suitable for optimisation, and methods exploiting this observation are presented in chapter 5.
	www.physics.uc.edu /~pkent/thesis/pkthnode20.html (1006 words)

	The Action Variational Principle in Cosmology
		During the last decade, the action variational principle of Peebles has proven to be a very applicable tool in the study of the formation of large-scale structures in the immediate neighbourhood of the Milky Way.
		It provides us with the mean to recreate the orbits of the galaxies from epochs early in the history of the Universe until the present, which in turn opens for a closer study of a number of subjects concerning the evolution of the Universe.
		In practise, the use of the action variational principle is a question of numerical optimization, a side of the method which previously has not been considered in detail.
	trond.hjorteland.com /thesis/hoved.html (395 words)

	A Variational Principle to the Kramers Equations with unbounded external forces - Huang (ResearchIndex) (Site not responding. Last check: 2007-11-04)
		The variational scheme is based on the idea of maximizing the entropy with respect to the Kantorovich functional associated with a certain cost function.
		42.0%: A Variational Principle for the Kramers Equation with Unbounded..
		Huang, C., `A variational principle to the Kramers equations with unbounded external forces', to appear in J. Math.
	citeseer.ist.psu.edu /191987.html (491 words)

	The Variational Principle and some applications
		There are very many examples of the use of the variational principle, several of which are suitable for further reading or for projects.
		Subsequent developments of the variational principle increased the number of parameters to get better fits to experiment, starting in the case of Helium with a paper by Hylleraas in 1930, with 6 terms, and culminating in 1962 with Pekeris who used 1078 terms, getting closer to the experimental result in the process, not always monotonically.
		This is a consistent feature of the variational principle in practice; more terms always give lower values, and hence, if all the important components are present in the model, better agreement with experiment.
	venables.asu.edu /quant/varprin.html (778 words)

	4aEA2 Variational principle modeling of class IV flextensional (Site not responding. Last check: 2007-11-04)
		The variational principle is an approximation method that allows one to obtain accurate estimates of certain quantities using relatively crude trial functions for the physical behavior.
		This principle is applied to transducer analysis by coupling a variational principle developed for the driving element (including piezoelectric effects) to one for the shell.
		The motion of the transducer is, in turn, coupled to a variational principle for the surface pressure in a fluid medium.
	www.auditory.org /asamtgs/asa93dnv/4aEA/4aEA2.html (184 words)

	Variational Programming
		The name is borrow from physics, where Hamilton's variational principle states that a physical trajectory of a particle is given by a path along which the action is minimal.
		As is the case in physics, the least action principle is a global one, not a local one.
		The goal of variational programming is: pragmatic simplicity, avoiding the rigidity of an approach that is too narrow, as well as the complexity of an approach that is too baroque and general-purpose.
	www.artcompsci.org /vol_1/v1_web/node4.html (1046 words)

	2.9 Fermat’s principle for light rays
		To set up a variational principle, we have to choose the trial curves among which the solution curves are to be determined and the functional that has to be extremized.
		Also for Fermat’s principle in ordinary optics, the extremum is never a local maximum, as is mentioned, e.g., in Born and Wolf [35], p.
		While Kovner’s principle, like the classical Fermat principle, is a varional principle for rays, the Frittelli-Newman principle is a variational principle for wave fronts.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-9/articlesu9.html (923 words)

	Variational principle and the Anti de Sitter/Conformal field theory (AdS/CFT) correspondence (Site not responding. Last check: 2007-11-04)
		We discuss how the variational principle can be used as a criterion for choosing, among scalar field actions implying the same equation of motion, the appropriate one for the AdS/CFT correspondence.
		Let us now interpret this result in the light of the ideas of reference [7] of appropriately using the variational principle to take into account the effects of the presence of the boundary.
		Concluding, the variational principle serves as a guide for choosing the appropriate action for the AdS/CFT correspondence, as explained in more detail in [9].
	www.sbf1.if.usp.br /eventos/enfpc/xxi/procs/res17 (954 words)

	On a Variational Principle for the Drag in Linear Hydrodynamics
		On a Variational Principle for the Drag in Linear Hydrodynamics
		The derivation of the set of equations for the induced force moments is given explicitly for two hydrodynamically relevant problems, namely a sphere moving slowly along the axis of a rotating viscous fluid and a sphere in Oseen flow.
		Moreover, the variational scheme is employed to study the influence of momentum convection on the drag on a sphere for small Reynolds and Taylor numbers.
	epubs.siam.org /sam-bin/dbq/article/29455 (219 words)

	[No title]
		The results of two recent articles expanding the Gibbs variational principle to encompass all of statistical mechanics, in which the role of external sources is made explicit, are utilized to further explicate the theory.
		Although Gibbs was silent on exactly why he chose this variational principle, his intent was quite clear: to define and construct a description of the equilibrium state.
		A variable, and therefore the system itself, is said to be thermally driven if no new variables other than those constrained experimentally are needed to characterize the resulting state, and if the Lagrange multipliers corresponding to variables other than those specified remain constant.
	w3.uwyo.edu /~wtg/applications_uni.html (6336 words)

	QMMS Lecture #6 (Venables/Heggie)
		Due to the central role this principle plays in QM, and given its frequent appearance during the rest of this course, it is worthwhile to review it at this stage.
		The variational principle is important not only because it tells us that an approximate wave function always gives an energy higher than the exact one, but it also tells us how to improve our approximate wave functions.
		Demonstrate the variational principle for ground states, by expanding the approximate wave function as a linear combination of the exact wave functions of the system.
	www.hpc.susx.ac.uk /~venables/qmms6.html (2351 words)

	Citations: The Variational Principles of Mechanics - Lanczos (ResearchIndex) (Site not responding. Last check: 2007-11-04)
		We consider these in the context of the eikonal equation, which arises in geometrical optics and has become of great interest for problems in computer vision [10, 22] It is the basis for continuous versions of mathematical morphology [9, 45, 64] as well as for Blum s grass re transform [5]....
		Lanczos, The variational principles of mechanics, (Dover, 1970).
		Calculus of variations is concerned with variations of functionals, where a functional is defined as some form of correspondence between a function and the set of the real numbers.
	citeseer.ist.psu.edu /context/8749/0 (2358 words)

	Variational Theory and the Variational Principle
		A very useful approximation method is known as the variational method.
		This is the basis of much of quantum chemistry, including Hartree-Fock theory, density functional theory, as well as variational quantum Monte Carlo.
		Although possible, in principle, this is very difficult to implement in practice unless the dimensionality of the system is very low.
	www.nyu.edu /classes/tuckerman/quant.mech/lectures/lecture_3/node1.html (237 words)

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