## Mathematical Methods for the Magnetohydrodynamics of Liquid Metals

*Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.001.0001
- Subject:
- Mathematics, Mathematical Physics

This text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis on the magnetohydrodynamics of liquid metals, on two-fluid flows, and ... 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

## NUMERICAL SIMULATION OF TWO-FLUID PROBLEMS

*Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre*

### in Mathematical Methods for the Magnetohydrodynamics of Liquid Metals

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.003.0005
- Subject:
- Mathematics, Mathematical Physics

This chapter deals with the discretization issues raised by multifluid magnetohydrodynamics problems. The additional difficulty compared to those in Chapter 3, is the presence of one (or many) free ... More

## ADVECTION AND ADVECTION DIFFUSION

*George Em Karniadakis and Spencer J. Sherwin*

### in Spectral/hp Element Methods for Computational Fluid Dynamics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528692
- eISBN:
- 9780191713491
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528692.003.0006
- Subject:
- Mathematics, Numerical Analysis

This chapter focuses on the scalar advection equation and develops a Galerkin discretization using the techniques described in Chapter 4. It includes an extended presentation of the discontinuous ... More

## Extremum principles

*Mauro Fabrizio and Morro Angelo*

### in Electromagnetism of Continuous Media: Mathematical Modelling and Applications

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198527008
- eISBN:
- 9780191713316
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527008.003.0009
- Subject:
- Mathematics, Applied Mathematics

This chapter shows various ways in which variational formulations can be established. Particular attention is given to the inverse problem, namely, to the determination of a variational formulation, ... More

## Competitive Economies

*Lars Peter Hansen and Thomas J. Sargent*

### in Recursive Models of Dynamic Linear Economies

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691042770
- eISBN:
- 9781400848188
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691042770.003.0007
- Subject:
- Economics and Finance, History of Economic Thought

This chapter describes a decentralized economy. It assigns ownership and decision making to three distinct economic entities, a household and two kinds of firms. It defines a competitive equilibrium. ... More

## Tonelli Lagrangians and Hamiltonians on Compact Manifolds

*Alfonso Sorrentino*

### in Action-minimizing Methods in Hamiltonian Dynamics (MN-50): An Introduction to Aubry-Mather Theory

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.003.0001
- Subject:
- Mathematics, Applied Mathematics

This chapter introduces the basic setting: Tonelli Lagrangians and Hamiltonians on a compact manifold. It discusses their main properties and some examples, and provides the opportunity to recall ... More

## From KAM Theory to Aubry-Mather Theory

*Alfonso Sorrentino*

### in Action-minimizing Methods in Hamiltonian Dynamics (MN-50): An Introduction to Aubry-Mather Theory

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.003.0002
- Subject:
- Mathematics, Applied Mathematics

This chapter discusses an illustrative example, namely the properties of invariant probability measures and orbits on KAM tori (or more generally, on invariant Lagrangian graphs). This will prepare ... More

## Lagrangian Mechanics with Time as a Coordinate

*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/9780198567264.003.0011
- Subject:
- Physics, Atomic, Laser, and Optical Physics

In traditional analytical mechanics, it is assumed that the system traces out a path (one-dimensional curved line) in a D dimensional configuration space that is defined by writing each of the ... More

## The Conceptual Framework of Quantum Field Theory

*Anthony Duncan*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The book attempts to provide an introduction to quantum field theory emphasizing conceptual issues frequently neglected in more ‘utilitarian’ treatments of the subject. The book is divided into four ... More

## Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

*S. G. Rajeev*

- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199670857
- eISBN:
- 9780191775154
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199670857.001.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This book begins with the ancient parts of classical mechanics: the variational principle, Lagrangian and Hamiltonian formalisms, and Poisson brackets. The simple pendulum provides a glimpse of the ... More

## Introduction to beam physics

*J. B. Rosenzweig*

### in Fundamentals of Beam Physics

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198525547
- eISBN:
- 9780191711725
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525547.003.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter begins by introducing particle accelerators in their scientific and historical context. It reviews methods in Lagrangian and Hamiltonian dynamics as well as special relativity in a ... More

## Fluid Mechanics and the SPH Method: Theory and Applications

*Damien Violeau*

- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199655526
- eISBN:
- 9780191741227
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655526.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials

This book aims at presenting the SPH method for fluid modelling from a theoretical and applied viewpoint. It comprises two parts that refer to each other. The first, dealing with the fundamentals of ... More

## Lagrangian Theory of Constraints

*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/9780198567264.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics

One attractive feature of the Lagrangian method is the ease with which it solves so-called constraint problems. This chapter presents several different ways of solving such problems, with examples of ... More

## Small Vibrations About Equilibrium

*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/9780198567264.003.0010
- Subject:
- Physics, Atomic, Laser, and Optical Physics

A number of interesting mechanical systems have one or more essentially stable equilibrium configurations. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called ... More

## Hamilton’s Principle and Noether’s Theorem

*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/9780198567264.003.0013
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter presents extended forms of Hamilton’s principle and the phase space Hamilton’s principle based on the extended Lagrangian and Hamiltonian methods developed earlier. It also discusses ... More

## Particle Motion in Curved Spacetime

*Valeri P. Frolov and Andrei Zelnikov*

### in Introduction to Black Hole Physics

- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199692293
- eISBN:
- 9780191731860
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199692293.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Particle motion in a curved spacetime is considered. The Lagrangian and Hamiltonian form of the equation of motion of a relativistic particle are described. We discuss here also the Hamilton‐Jacobi ... More

## Lattice gauge theory beyond the Standard Model

*Thomas Appelquist and Ethan T. Neil*

### in Modern Perspectives in Lattice QCD: Quantum Field Theory and High Performance Computing: Lecture Notes of the Les Houches Summer School: Volume 93, August 2009

- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199691609
- eISBN:
- 9780191731792
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199691609.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter gives an overview of methods and recent studies in the application of lattice gauge theory to physics beyond the standard model. It focuses on theories in which electroweak symmetry is ... More

## Lagrangian and Hamiltonian mechanics

*D. Violeau*

### in Fluid Mechanics and the SPH Method: Theory and Applications

- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199655526
- eISBN:
- 9780191741227
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655526.003.0001
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter details the basics of Lagrangian and Hamiltonian mechanics. From the basic principles of mechanics (Galilean invariance, least action principle), the chapter builds the fundamental ... More

## The Lazy Universe: An Introduction to the Principle of Least Action

*Jennifer Coopersmith*

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198743040
- eISBN:
- 9780191802966
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198743040.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, History of Physics

Action and the Principle of Least Action are explained: what Action is, why the Principle of Least Action works, why it underlies all physics, and what are the insights gained into energy, space, and ... More