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Integral principles

JESPER LÜTZEN

in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

Published in print:
2005
Published Online:
January 2010
ISBN:
9780198567370
eISBN:
9780191717925
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567370.003.0021
Subject:
Physics, History of Physics

In his book Principles of Mechanics, Heinrich Hertz treats integral principles in three parts. First, he discusses the properties of geodesics and their relation to straightest paths in a purely ... More


The View from General Relativity

Harvey R. Brown

in Physical Relativity: Space-time structure from a dynamical perspective

Published in print:
2005
Published Online:
September 2006
ISBN:
9780199275830
eISBN:
9780191603914
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0199275831.003.0009
Subject:
Philosophy, Philosophy of Science

In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the ... More


RADON TRANSFORM ON GROUPS

Leon Ehrenpreis

in The Universality of the Radon Transform

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198509783
eISBN:
9780191709166
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509783.003.0007
Subject:
Mathematics, Mathematical Physics

This chapter applies the book's methods to various Lie groups. In particular, to the horocyclic and geodesic transforms on G/K.


Lorentz Geometry

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

Published in print:
2008
Published Online:
May 2009
ISBN:
9780199230723
eISBN:
9780191710872
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230723.003.0001
Subject:
Mathematics, Applied Mathematics

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local ... More


Hyperbolic Dynamics and Brownian Motion: An Introduction

Jacques Franchi and Yves Le Jan

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199654109
eISBN:
9780191745676
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199654109.001.0001
Subject:
Mathematics, Mathematical Physics

The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) ... More


The Built Environment: Buckminster Fuller's Spaceship Earth

Linda Sargent Wood

in A More Perfect Union: Holistic Worldviews and the Transformation of American Culture after World War II

Published in print:
2010
Published Online:
September 2010
ISBN:
9780195377743
eISBN:
9780199869404
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195377743.003.0003
Subject:
History, American History: 20th Century

Structural engineer Buckminster Fuller's geodesic dome was a visible manifestation of holism. Held together in what he described as “synergetic” wholeness, each part relying on neighboring parts for ... More


On Relativity, Time Reckoning, and the Topology of Time Series

Roberto Torretti

in The Arguments of Time

Published in print:
2006
Published Online:
January 2012
ISBN:
9780197263464
eISBN:
9780191734748
Item type:
chapter
Publisher:
British Academy
DOI:
10.5871/bacad/9780197263464.003.0004
Subject:
Philosophy, General

This chapter devotes equal attention to special relativity and general relativity. It first describes the history of the analysis of distant simultaneity, up to and including Einstein's procedure in ... More


The geometric approach to GWs

Michele Maggiore

in Gravitational Waves: Volume 1: Theory and Experiments

Published in print:
2007
Published Online:
January 2008
ISBN:
9780198570745
eISBN:
9780191717666
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570745.003.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses how gravitational waves emerge from general relativity, and what their properties are. The most straightforward approach is ‘linearized theory’, where the Einstein equations ... 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


Relativity, Gravitation and Cosmology: A Basic Introduction

Ta-Pei Cheng

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199573639
eISBN:
9780191722448
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199573639.001.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, the principle of equivalence, Riemannian ... More


Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Christopher D. Sogge

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.001.0001
Subject:
Mathematics, Numerical Analysis

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ... More


Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Jean-Michel Bismut

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691151298
eISBN:
9781400840571
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691151298.001.0001
Subject:
Mathematics, Geometry / Topology

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators ... More


Metric description of a curved space

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology: A Basic Introduction

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199573639
eISBN:
9780191722448
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199573639.003.0005
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. ... More


GR as a geometric theory of gravity – I

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology: A Basic Introduction

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199573639
eISBN:
9780191722448
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199573639.003.0006
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

A geometric description of equivalence principle physics of gravitational time dilation is presented. In this geometric theory, the metric plays the role of relativistic gravitational potential. ... More


The three body problem

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/9780199670857.003.0011
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter derives the consequences of scale invariance in the three body problem. Jacobi co-ordinates are introduced. The orbits are geodesics in a certain metric. The three body problem with ... More


GR as a geometric theory of gravity — II

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology: A Basic Introduction

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199573639
eISBN:
9780191722448
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199573639.003.0014
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

The mathematical realization of equivalence principle (EP) is the principle of general covariance. General relativity (GR) equations must be covariant with respect to general coordinate ... More


Mathematicians on the geometrization of the Hamilton–Jacobi formalism

JESPER LÜTZEN

in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

Published in print:
2005
Published Online:
January 2010
ISBN:
9780198567370
eISBN:
9780191717925
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567370.003.0024
Subject:
Physics, History of Physics

It has long since been remarked by mathematicians that William Rowan Hamilton's method contains purely geometrical truths, and that a peculiar mode of expression, suitable to it, is required in order ... More


Cosmology

Carlo Giunti and Chung W. Kim

in Fundamentals of Neutrino Physics and Astrophysics

Published in print:
2007
Published Online:
January 2010
ISBN:
9780198508717
eISBN:
9780191708862
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198508717.003.0016
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter introduces the standard cosmological model and explains basic general relativity, Robertson–Walker metric (geodesic motion, redshift, Hubble's law, angular diameter-redshift relation, ... 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


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


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