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Sergey N. Dorogovtsev

in Lectures on Complex Networks

Published in print:
2010
Published Online:
May 2010
ISBN:
9780199548927
eISBN:
9780191720574
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199548927.003.0015
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this concluding section of the text, the three major milestones marking the history of the exploration of networks are indicated. These are: Leonhard Euler's work (1735), the introduction of ... More


RELATIVISTIC HYDRODYNAMICS

Miguel Alcubierre

in Introduction to 3+1 Numerical Relativity

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199205677
eISBN:
9780191709371
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199205677.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter presents the derivation of dynamical equations governing the motion of a perfect fluid, known as the Euler equations, both in the special and general relativistic cases. It considers ... More


Fluid Dynamics: Part 1: Classical Fluid Dynamics

Anatoly I. Ruban and Jitesh S. B. Gajjar

Published in print:
2014
Published Online:
August 2014
ISBN:
9780199681730
eISBN:
9780191761607
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199681730.001.0001
Subject:
Physics, Soft Matter / Biological Physics

This book, the first of a four-part series on fluid dynamics, consists of four chapters on classical theory suitable for an introductory undergraduate course. Chapter 1 discusses the continuum ... More


The rigid body

S. G. Rajeev

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

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199670857
eISBN:
9780191775154
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670857.003.0005
Subject:
Physics, Atomic, Laser, and Optical Physics

In this chapter, the moment of inertia is defined and Euler's equations for the rigid body are derived. They can be solved in terms of Jacobi elliptic functions.


Hamilton–Jacobi theory

S. G. Rajeev

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

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199670857
eISBN:
9780191775154
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670857.003.0009
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter derives the Hamilton-Jacobi equation. Euler's two centre problem is solved by using separation of variables in the elliptical co-ordinate system. The connection to the eikonal equation ... More


The Consequences of Extensivity

Robert H. Swendsen

in An Introduction to Statistical Mechanics and Thermodynamics

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199646944
eISBN:
9780191775123
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646944.003.0013
Subject:
Physics, Condensed Matter Physics / Materials

Although extensivity is not an essential property of thermodynamic systems, it is a very useful concept for analysing the properties of materials when we can ignore surface and boundary effects. This ... More


Equilibrium and Non-Equilibrium Collisional Regimes

Raymond Brun

in Introduction to Reactive Gas Dynamics

Published in print:
2009
Published Online:
May 2009
ISBN:
9780199552689
eISBN:
9780191720277
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199552689.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter defines collisional regimes of gaseous flows, including equilibrium and non-equilibrium situations which are described with Euler equations. It examines monatomic and diatomic gas flows, ... More


Nonlinear Bending of Straight Beams

J. N. REDDY

in An Introduction to Nonlinear Finite Element Analysis

Published in print:
2004
Published Online:
January 2010
ISBN:
9780198525295
eISBN:
9780191711671
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198525295.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter considers the application of the finite element method to the analysis of a slightly more complicated one-dimensional nonlinear problem: the bending of straight beams. When the applied ... More


The Stochastic Simulation Method

Heinz-Peter Breuer and Francesco Petruccione

in The Theory of Open Quantum Systems

Published in print:
2007
Published Online:
January 2010
ISBN:
9780199213900
eISBN:
9780191706349
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213900.003.07
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The formulation of the dynamics of open quantum systems by means of stochastic processes in Hilbert space leads to efficient Monte–Carlo simulation techniques which are introduced and examined in ... More


Kinematics of Rotation

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/9780198567264.003.0008
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter deals with techniques needed to define the location and orientation of a moving rigid body. Rigid bodies are characterised and the center of mass of a rigid body is discussed along with ... More


Rotational Dynamics

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/9780198567264.003.0009
Subject:
Physics, Atomic, Laser, and Optical Physics

The successful description of the motion of a rigid body is one of the triumphs of Newtonian mechanics. Having learned in the previous chapter how to specify the position and orientation of a rigid ... More


ROTATIONAL DIFFUSION

Robert M. Mazo

in Brownian Motion: Fluctuations, Dynamics, and Applications

Published in print:
2008
Published Online:
January 2010
ISBN:
9780199556441
eISBN:
9780191705625
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199556441.003.0015
Subject:
Physics, Condensed Matter Physics / Materials

This chapter treats rotational diffusion in the Smoluchowski limit, but not limited to uniaxial diffusion. The rotational Langevin equation for a spherical particle with a homogeneous external field ... More


Mathematical Tools

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

Published in print:
2019
Published Online:
September 2019
ISBN:
9780199678839
eISBN:
9780191878589
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199678839.003.0019
Subject:
Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from the Bethe ansatz equations for a Heisenberg quantum spin chain of finite length. It discusses how the ... More


Fluid Mechanics: A Geometrical Point of View

S. G. Rajeev

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.001.0001
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using ... More


Point Transformations in Lagrangian Mechanics

Peter Mann

in Lagrangian and Hamiltonian Dynamics

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198822370
eISBN:
9780191861253
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198822370.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses point transformations in Lagrangian mechanics. Sometimes, when solving problems, it is useful to change coordinates in velocity phase space to better suit and simplify the ... More


Differential Equations

Peter Mann

in Lagrangian and Hamiltonian Dynamics

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198822370
eISBN:
9780191861253
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198822370.003.0035
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used ... More


Geometric Integrators

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0015
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric ... More


Later years

I.S. Glass

in Nicolas-Louis De La Caille, Astronomer and Geodesist: South African Astronomical Observatory

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199668403
eISBN:
9780191749315
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199668403.003.0006
Subject:
Physics, History of Physics

La Caille returned to France via Mauritius and Reunion. He was received with acclaim at the Academy, where he gave several lectures on his Cape experiences. He published a map of the Cape which ... More


Lagrangian Mechanics

David D. Nolte

in Introduction to Modern Dynamics: Chaos, Networks, Space, and Time

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198844624
eISBN:
9780191880216
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198844624.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Dynamical systems follow trajectories for which the mechanical action integrated along the trajectory is an extremum. The action is defined as the time average of the difference between kinetic and ... More


Introduction to Differential Equations

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199566433
eISBN:
9780191774966
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199566433.003.0029
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on differential equations, dynamical systems that change continuously, with the variable of interest having a value at every instant. It first provides an overview of discrete ... More


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