## 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

## 1 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

## 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

## Second Order and Gaussian Fields

*Ulf Grenander and Michael I. Miller*

### in Pattern Theory: From representation to inference

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198505709
- eISBN:
- 9780191916564
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198505709.003.0010
- Subject:
- Computer Science, Programming Languages

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in ... 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

## 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

## 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

## 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

## 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

## 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

## 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