## 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