Some Remarks on the Use of the Variational Principle for the Second Order Energy
Approximate ground state wave function used in variational principle for second order energy
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Some remarks on the use of the variational principle for the second-order energy.
Second order energy correction calculated by Hylleraas variation principle with approximate zero order function, shows no relationship to true value
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Electromagnetic axisymmetric finite elements based on a gauged four-potential variational principle
Electromagnetic finite elements are derived based on a variational principle that uses the electromagnetic four-potential as a primary variable. The Lorentz gage normalization is incorporated as a constraint condition through a Lagrange multiplier field to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of high-temperature superconductor devices with aerospace applications. The main advantages are: jump discontinuities on interfaces are naturally handled; no a priori approximations are invoked; and the number of degrees of freedom per node remains modest as the problem dimensionality increases. The new elements are tested on two magnetostatic axisymmetric problems. The results are in excellent agreement with analytical solutions and previous solutions for the 1D problem of a conducting infinite wire, in which case the multiplier field has no effect. For materials of widely different permeability, jump conditions are naturally accommodated by the present formulation.
Schuler, J.↗
Nonparametric solutions to the variational principle of ideal magnetohydrodynamics
In an effort to gain a better understanding of MHD equilibria in three dimensions, the lower dimensional cases are studied. The solution of the three-dimensional problem is based on the classical variational principle of ideal magnetohydrodynamics. The crucial assumption for the numerical method is the existence of a nested set of toroidal flux surfaces, which is then used as a coordinate. This paper studies the nonparametric solutions to this variational problem in those cases when the direct solution is known to have islands. A form of the variational principle for the slab geometry is described; the one-dimensional problem is analyzed; and asymptotic expansions and numerical solutions to the two-dimensional problem are discussed. An example is presented which shows that the assumption of nested flux surfaces need not rule out the occurrence of islands.
Betancourt, O. L.↗
Two notes on perturbation theory. 1 - Interchange theorems and the variation principle. 2 - PV in a different notation
Interchange theorems and variational principle in perturbation theory
Epstein, S. T.↗
Green's formula and variational principles for cosmic-ray transport with application to rotating and shearing flows
Green's theorem and Green's formula for the diffusive cosmic-ray transport equation in relativistic flows are derived. Green's formula gives the solution of the transport equation in terms of the Green's function of the adjoint transport equation, and in terms of distributed sources throughout the region R of interest, plus terms involving the particle intensity and streaming on the boundary. The adjoint transport equation describes the time-reversed particle transport. An Euler-Lagrange variational principle is then obtained for both the mean scattering frame distribution function f, and its adjoint f(dagger). Variations of the variational functional with respect to f(dagger) yield the transport equation, whereas variations of f yield the adjoint transport equation. The variational principle, when combined with Noether's theorem, yields the conservation law associated with Green's theorem. An investigation of the transport equation for steady, azimuthal, rotating flows suggests the introduction of a new independent variable H to replace the comoving frame momentum variable p'. For the case of rigid rotating flows, H is conserved and is shown to be analogous to the Hamiltonian for a bead on a rigidly rotating wire. The variable H corresponds to a balance between the centrifugal force and the particle inertia in the rotating frame. The physical interpretation of H includes a discussion of nonrelativistic and special relativistic rotating flows as well as the cases of aziuthal, differentially rotating flows about Schwarzs-child and Kerr black holes. Green's formula is then applied to the problem of the acceleration of ultra-high-energy cosmic rays by galactic rotation. The model for galactic rotation assumes an angular velocity law Omega = Omega(sub 0)(omega(sub 0)/omega), where omega denotes radial distance from the axis of rotation. Green's functions for the galactic rotation problem are used to investigate the spectrum of accelerated particles arising from monoenergetic and truncated power-law sources. We conclude that it is possible to accelerate particles beyond the knee by galactic rotation, but not in sufficient number to adequately explain the observed spectrum.
Webb, G. M.↗
Complementary variational principle and duality in mathematical programming.
The relationship between the complementary variational principle and duality in mathematical programming is demonstrated through a geometric approach in a Hilbert space setting. A necessary and sufficient condition for the existence of such a principle is given in the case of a convex functional constrained by linear dynamics. Its relationship to the Kuhn-Tucker saddle point theory is indicated. Applications to various programming and control problems are discussed.
Chan, W. L.↗
Applications of variational principles in computing rotational flows
Ecer and Akay (1983) have developed a variational formulation of rotational flow for Euler equations. The present paper provides a summary of these developments. The considered variational formulation provides a transformation of a type considered by Clebsch (1859). In this transformation, a new set of variables replaces the more commonly used primitive variables u(i), rho and p. Here, u(i) denotes the velocity components, while rho is the density, and p the pressure. The employed transformation produces a natural uncoupling of the equations when written in a quasi-linear form. After obtaining the governing equations in terms of the 'Clebsch variables', a solution scheme developed for calculating steady flows is discussed. Attention is given to numerical solutions of Euler equations based on the derived variational principles, and a study of inviscid, separated flows is conducted.
Ecer, A.↗
A variational principle for compressible fluid mechanics: Discussion of the multi-dimensional theory
The variational principle for compressible fluid mechanics previously introduced is extended to two dimensional flow. The analysis is stable, exactly conservative, adaptable to coarse or fine grids, and very fast. Solutions for two dimensional problems are included. The excellent behavior and results lend further credence to the variational concept and its applicability to the numerical analysis of complex flow fields.
Prozan, R. J.↗
A variational principle for magnetohydrodynamic channel flow
MHD flow in channel of arbitrary cross section, deriving velocity and electric potential distribution by variational principle
Wenger, N. C.↗
Superconducting axisymmetric finite elements based on a gauged potential variational principle. Part 1: Formulation
The present work is part of a research program for the numerical simulation of electromagnetic (EM) fields within conventional Ginzburg-Landau (GL) superconductors. The final goal of this research is to formulate, develop and validate finite element (FE) models that can accurately capture electromagnetic thermal and material phase changes in a superconductor. The formulations presented here are for a time-independent Ginzburg-Landau superconductor and are derived from a potential-based variational principle. We develop an appropriate variational formulation of time-independent supercontivity for the general three-dimensional case and specialize it to the one-dimensional case. Also developed are expressions for the material-dependent parameters alpha and beta of GL theory and their dependence upon the temperature T. The one-dimensional formulation is then discretized for finite element purposes and the first variation of these equations is obtained. The resultant Euler equations contain nonlinear terms in the primary variables. To solve these equations, an incremental-iterative solution method is used. Expressions for the internal force vector, external force vector, loading vector and tangent stiffness matrix are therefore developed for use with the solution procedure.
Schuler, James J.↗
A variational principle for compressible fluid mechanics. Discussion of the one-dimensional theory
The second law of thermodynamics is used as a variational statement to derive a numerical procedure to satisfy the governing equations of motion. The procedure, based on numerical experimentation, appears to be stable provided the CFL condition is satisfied. This stability is manifested no matter how severe the gradients (compression or expansion) are in the flow field. For reasons of simplicity only one dimensional inviscid compressible unsteady flow is discussed here; however, the concepts and techniques are not restricted to one dimension nor are they restricted to inviscid non-reacting flow. The solution here is explicit in time. Further study is required to determine the impact of the variational principle on implicit algorithms.
Prozan, R. J.↗
Variational formulation of high performance finite elements: Parametrized variational principles
High performance elements are simple finite elements constructed to deliver engineering accuracy with coarse arbitrary grids. This is part of a series on the variational basis of high-performance elements, with emphasis on those constructed with the free formulation (FF) and assumed natural strain (ANS) methods. Parametrized variational principles that provide a foundation for the FF and ANS methods, as well as for a combination of both are presented.
Felippa, Carlos A.↗
Variational formulation of high-performance finite elements - Parametrized variational principles
High-performance (HP) elements are simple finite elements constructed to deliver engineering accuracy with coarse arbitrary grids. This paper is part of a series on the variational basis of HP elements, with emphasis on those constructed with the free formulation (FF) and assumed natural strain (ANS) methods. The present paper studies parametrized variational principles that provide a foundation for the FF and ANS methods, as well as for a combustion of both methods.
Felippa, C. A.↗
Improved techniques for outgoing wave variational principle calculations of converged state-to-state transition probabilities for chemical reactions
Improved techniques and well-optimized basis sets are presented for application of the outgoing wave variational principle to calculate converged quantum mechanical reaction probabilities. They are illustrated with calculations for the reactions D + H2 yields HD + H with total angular momentum J = 3 and F + H2 yields HF + H with J = 0 and 3. The optimization involves the choice of distortion potential, the grid for calculating half-integrated Green's functions, the placement, width, and number of primitive distributed Gaussians, and the computationally most efficient partition between dynamically adapted and primitive basis functions. Benchmark calculations with 224-1064 channels are presented.
Mielke, Steven L.↗
Virial theorem, the variational principle, and nonlinear parameters in quantum mechanics - Application to the 3 super 1 D and 3 super 3 D states of the helium isoelectronic series.
Nonlinear parameters in many body quantum mechanical system, discussing virial theorem, variational principle and application to various states of He isoelectronic series
Brown, R. T.↗
An MHD variational principle that admits reconnection
The variational approach of Pfirsch and Sudan's averaged magnetohydrodynamics (MHD) to the stability of a line-tied current layer is summarized. The effect of line-tying on current sheets that might arise in line-tied magnetic flux tubes by estimating the growth rates of a resistive instability using a variational method. The results show that this method provides a potentially new technique to gauge the stability of nearly ideal magnetohydrodynamic systems. The primary implication for the stability of solar coronal structures is that tearing modes are probably constant at work removing magnetic shear from the solar corona.
Rilee, M. L.↗
Variational principle for a prototype Rastall theory of gravitation
A prototype of Rastall's theory of gravity, in which the divergence of the energy-momentum tensor is proportional to the gradient of the scalar curvature, is shown to be derivable from a variational principle. Both the proportionality factor and the unrenormalized gravitational constant are found to be covariantly constant, but not necessarily constant. The prototype theory is, therefore, a gravitational theory with variable gravitational constant.
Smalley, L. L.↗