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Variational Principles

Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153551
eISBN:
9781400842698
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153551.003.0007
Subject:
Mathematics, Analysis

This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where ... More


Hamilton’s Principle

Oliver Johns

in Analytical Mechanics for Relativity and Quantum Mechanics

Published in print:
2005
Published Online:
January 2010
ISBN:
9780198567264
eISBN:
9780191717987
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567264.003.0006
Subject:
Physics, Atomic, Laser, and Optical Physics

The general calculus of variations discussed in the previous chapter may be used to derive variational principles in mechanics. Two different, but closely related, variational principles are ... More


Convolutional Variational Principles

Józef Ignaczak and Martin Ostoja‐Starzewski

in Thermoelasticity with Finite Wave Speeds

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199541645
eISBN:
9780191716164
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199541645.003.0005
Subject:
Mathematics, Applied Mathematics, Mathematical Physics

This chapter first presents the alternative descriptions of a conventional thermoelastic process in the Green‐Lindsay theory, and then gives the variational principles for a conventional and ... More


The Hartree–Fock approximation

Helmut Hofmann

in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

Published in print:
2008
Published Online:
September 2008
ISBN:
9780198504016
eISBN:
9780191708480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504016.003.0020
Subject:
Physics, Nuclear and Plasma Physics

This chapter formulates the Hartree-Fock approximation with density matrices, and discusses the properties of the Hartree-Fock equations together with the energy functional for the ground state ... More


FORMAL TRANSPORT THEORY

J. M. Ziman

in Electrons and Phonons: The Theory of Transport Phenomena in Solids

Published in print:
2001
Published Online:
September 2007
ISBN:
9780198507796
eISBN:
9780191709937
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507796.003.0007
Subject:
Physics, Condensed Matter Physics / Materials

The analysis of the microscopic processes by which electrons and phonons are scattered is only half the calculation of the macroscopic transport coefficients. To discuss the flow of electric and ... More


Fr échet Differentiability of Real-Valued Functions

Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153551
eISBN:
9781400842698
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153551.003.0012
Subject:
Mathematics, Analysis

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the ... More


TRANSPORT PHENOMENA IN A MAGNETIC FIELD

J. M. Ziman

in Electrons and Phonons: The Theory of Transport Phenomena in Solids

Published in print:
2001
Published Online:
September 2007
ISBN:
9780198507796
eISBN:
9780191709937
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507796.003.0012
Subject:
Physics, Condensed Matter Physics / Materials

This chapter discusses the transport properties of a solid in a magnetic field. Topics covered include the Hall effect, magnetoresistance, thermomagnetic phenomena, the method of Jones and Zener, ... More


Introduction

Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153551
eISBN:
9781400842698
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153551.003.0001
Subject:
Mathematics, Analysis

This book deals with the existence of Fréchet derivatives of Lipschitz functions from X to Y, where X is an Asplund space and Y has the Radon-Nikodým property (RNP). It considers whether every ... More


Exploiting the Variational Principle

Mike Finnis

in Interatomic Forces in Condensed Matter

Published in print:
2003
Published Online:
January 2010
ISBN:
9780198509776
eISBN:
9780191709180
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509776.003.0003
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter begins with a discussion of the ubiquitous theorem that underlies much of the theory of interatomic forces, namely the Hellmann–Feynman theorem, and its derivation from the variational ... More


Equilibrium Statistical Mechanics and Thermodynamics

T. Grandy Walter

in Entropy and the Time Evolution of Macroscopic Systems

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199546176
eISBN:
9780191720161
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199546176.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter applies the theoretical tolls developed in Chapter 3 to macroscopic systems in thermal equilibrium, wherein the Gibbs variational principle is understood as defining the equilibrium ... More


PARTITION FUNCTION AND SPECTRUM

Jean Zinn-Justin

in Path Integrals in Quantum Mechanics

Published in print:
2004
Published Online:
January 2010
ISBN:
9780198566748
eISBN:
9780191717994
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566748.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

A direct application of the calculation of the partition function is provided by determining the spectrum of a Hamiltonian (which is assumed to be discrete, for simplicity). In this chapter, only ... More


Hamilton's Principle

Oliver Davis Johns

in Analytical Mechanics for Relativity and Quantum Mechanics

Published in print:
2011
Published Online:
December 2013
ISBN:
9780191001628
eISBN:
9780191775161
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780191001628.003.0006
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter shows that the general calculus of variations can be used to derive variational principles in mechanics. Two different, but closely related, variational principles are presented here: ... More


Properties of the time-dependent xc potential

Carsten A. Ullrich

in Time-Dependent Density-Functional Theory: Concepts and Applications

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199563029
eISBN:
9780191775130
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199563029.003.0006
Subject:
Physics, Condensed Matter Physics / Materials

This chapter reviews the properties of the time-dependent exchange-correlation (xc) potential. It starts with a discussion of the meaning of the universal xc functional, and then covers a number of ... More


Relativistic Perfect Fluids

Luciano Rezzolla and Olindo Zanotti

in Relativistic Hydrodynamics

Published in print:
2013
Published Online:
January 2014
ISBN:
9780198528906
eISBN:
9780191746505
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528906.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter builds the mathematical framework that is behind relativistic perfect fluids, namely those fluids for which viscous effects and heat fluxes are zero. Starting from the definition of the ... More


The essential role of functional integrals in modern physics

Jean Zinn-Justin

in From Random Walks to Random Matrices

Published in print:
2019
Published Online:
August 2019
ISBN:
9780198787754
eISBN:
9780191829840
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198787754.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 3 illustrates by a number of examples the essential role of functional integrals in physics. For example, the path integral representation of quantum mechanics explains why many basic ... More


On the Shoulders of Giants

David D. Nolte

in Galileo Unbound: A Path Across Life, the Universe and Everything

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198805847
eISBN:
9780191843808
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805847.003.0004
Subject:
Physics, History of Physics

Galileo’s parabolic trajectory launched a new approach to physics that was taken up by a new generation of scientists like Isaac Newton, Robert Hooke and Edmund Halley. The English Newtonian ... More


Rotating Shallow-Water Models with Full Coriolis Force

Vladimir Zeitlin

in Geophysical Fluid Dynamics: Understanding (almost) everything with rotating shallow water models

Published in print:
2018
Published Online:
April 2018
ISBN:
9780198804338
eISBN:
9780191842627
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198804338.003.0016
Subject:
Physics, Geophysics, Atmospheric and Environmental Physics

The derivation of the rotating shallow-water model by vertical averaging is carried on in the tangent plane approximation without neglecting the vertical component of the Coriolis force, and ... More


Fundamental Ideas

Nicholas Manton and Nicholas Mee

in The Physical World: An Inspirational Tour of Fundamental Physics

Published in print:
2017
Published Online:
July 2017
ISBN:
9780198795933
eISBN:
9780191837111
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198795933.003.0002
Subject:
Physics, Condensed Matter Physics / Materials

Chapter 1 offers a simple introduction to the use of variational principles in physics. This approach to physics plays a key role in the book. The chapter starts with a look at how we might minimize ... More


The time-dependent optimized effective potential

Carsten A. Ullrich

in Time-Dependent Density-Functional Theory: Concepts and Applications

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199563029
eISBN:
9780191775130
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199563029.003.0011
Subject:
Physics, Condensed Matter Physics / Materials

The optimized effective potential (OEP) method is a formally exact approach to treat orbital-dependent exchange-correlation functionals. The first part of this chapter reviews the static OEP method. ... More


Augmented Plane Wave and Green’s Function Methods*

J. B. Ketterson

in The Physics of Solids

Published in print:
2016
Published Online:
December 2016
ISBN:
9780198742906
eISBN:
9780191821523
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198742906.003.0013
Subject:
Physics, Condensed Matter Physics / Materials

This chapter discusses two techniques for calculating wave functions and energy levels in solids: the augmented plane wave (APW) and Green’s function methods. The APW method is derived using a ... More


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