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## Branes on Poisson Varieties

*Marco Gualtieri*

### in The Many Facets of Geometry: A Tribute to Nigel Hitchin

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0018
- Subject:
- Mathematics, Geometry / Topology

This chapter takes a second look at a classical structure in differential and algebraic geometry — that of a holomorphic Poisson structure — which is a complex manifold with a holomorphic Poisson ... More

## Canonical Transformations

*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/9780198567264.003.0017
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter considers a broader class of transformations known as canonical transformations, which transform the whole of the extended phase space in a more general way. Each new canonical ... More

## Hamiltonian 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/9780198567264.003.0012
- Subject:
- Physics, Atomic, Laser, and Optical Physics

In the previous chapter, the traditional Lagrange equations were used as the basis for an extended Lagrangian theory in which time is treated as a coordinate. This extended theory combined the ... More

## Canonical Transformations

*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/9780191001628.003.0019
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter describes canonical transformations as the most general phase-space transformations that preserve the extended Hamilton equations. There are several equivalent definitions of canonical ... More

## Introduction To Hamiltonian Mechanics

*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/9780198567264.003.0004
- Subject:
- Physics, Atomic, Laser, and Optical Physics

The power of Lagrangian mechanics has caused generations of students to wonder why it is necessary or even desirable, to recast mechanics in Hamiltonian form. The answer is that the Hamiltonian ... More

## The Hamiltonian equations of motion. Poisson brackets

*Gleb L. Kotkin and Valeriy G. Serbo*

### in Exploring Classical Mechanics: A Collection of 350+ Solved Problems for Students, Lecturers, and Researchers - Second Revised and Enlarged English Edition

- Published in print:
- 2020
- Published Online:
- October 2020
- ISBN:
- 9780198853787
- eISBN:
- 9780191888236
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853787.003.0010
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle ... 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

## Hamilton’s principle

*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/9780199670857.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics

The variational principle is formulated in first order form, deriving Hamilton's equations. Poisson brackets are found as generating infinitesimal canonical transformations. Symmetries are found to ... More

## The Hamiltonian equations of motion. Poisson brackets

*Gleb L. Kotkin and Valeriy G. Serbo*

### in Exploring Classical Mechanics: A Collection of 350+ Solved Problems for Students, Lecturers, and Researchers - Second Revised and Enlarged English Edition

- Published in print:
- 2020
- Published Online:
- October 2020
- ISBN:
- 9780198853787
- eISBN:
- 9780191888236
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853787.003.0023
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle ... More

## Hamiltonian mechanics including constraints and fields

*Rodolfo Gambini and Jorge Pullin*

### in A First Course in Loop Quantum Gravity

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199590759
- eISBN:
- 9780191774980
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199590759.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses the formulation of mechanical systems in Hamiltonian form, and briefly describes the role that constraints among the canonical variables play as generators of gauge symmetries. ... More

## Continuous systems

*Tom Lancaster and Stephen J. Blundell*

### in Quantum Field Theory for the Gifted Amateur

- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199699322
- eISBN:
- 9780191779435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199699322.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter describes how the arguments in the previous part, which worked on discrete systems, can be generalized to the continuum limit. After reviewing Hamilton’s formulation of classical ... More

## Hamilton’s Equations & Routhian Reduction

*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/9780198822370.003.0016
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this chapter, the Poisson bracket and angular momentum are investigated and first integrals are used to develop conservation laws as a canonical Noether’s theorem. The Poisson bracket was ... More

## Topics in Mechanics

*David J. N. Limebeer and Matteo Massaro*

### in Dynamics and Optimal Control of Road Vehicles

- Published in print:
- 2018
- Published Online:
- April 2019
- ISBN:
- 9780198825715
- eISBN:
- 9780191864636
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198825715.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

Chapter 2 provides a comprehensive review of the classical mechanics required when building vehicle models. Both vector-based methods and the variational approach to classical mechanics are reviewed, ... More

## Absolute time

*George Jaroszkiewicz*

### in Images of Time: Mind, Science, Reality

- Published in print:
- 2016
- Published Online:
- January 2016
- ISBN:
- 9780198718062
- eISBN:
- 9780191787553
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198718062.003.0014
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses Newton’s concept of absolute time and its use in classical mechanics. It defines clocks and our freedom to change them, that is, reparametrization of time. The chapter goes on ... More

## Hamilton and Green

*Jennifer Coopersmith*

### in Energy, the Subtle Concept: The discovery of Feynman's blocks from Leibniz to Einstein

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198716747
- eISBN:
- 9780191800955
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198716747.003.0013
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, History of Physics

The unknown George Green, from Nottingham, self-published An Essay on the Mathematical Analysis of Electricity and Magnetism, in which the potential function, V, was defined for the first time. The ... More

## Basic Statistical Mechanics

*C. G. Gray and K. E. Gubbins*

### in Theory of Molecular Fluids: I: Fundamentals

- Published in print:
- 1984
- Published Online:
- November 2020
- ISBN:
- 9780198556022
- eISBN:
- 9780191919251
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198556022.003.0009
- Subject:
- Chemistry, Theoretical Chemistry

In this chapter we introduce distribution functions for molecular momenta and positions. All equilibrium properties of the system can be calculated if both the intermolecular potential energy and ... More

## Kinetic theory

*Nathalie Deruelle and Jean-Philippe Uzan*

### in Relativity in Modern Physics

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198786399
- eISBN:
- 9780191828669
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198786399.003.0010
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is ... More

## Review Of Some Mathematical And Physical Subjects

*Abraham Nitzan*

### in Title Pages

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198529798
- eISBN:
- 9780191916649
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198529798.003.0006
- Subject:
- Chemistry, Physical Chemistry

This chapter reviews some subjects in mathematics and physics that are used in different contexts throughout this book. The selection of subjects and the level of their coverage reflect the ... More

## Lagrangian Field Theory

*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/9780198822370.003.0025
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In this chapter, Hamiltonian field theory is derived classically via a Hamiltonian density, using the zeroth component of a 4-momentum density. In field theory, space and time are considered to be on ... More

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