J. B. Rosenzweig
- 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 ...
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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 unified way in order to build up the tools needed to examine the dynamics of charged particle beams. It notes that these general subjects gave way to describing the motion of charged particles in beams. It investigates the notion of phase space, and the conservation of its density in Hamiltonian systems — the Louiville theorem. It introduces the concept of the design trajectory that allows nearby trajectories to be defined. It adds that the design trajectory also gives one the freedom to analyse the motion using distance along such a trajectory as the independent variable, instead of time.Less
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 unified way in order to build up the tools needed to examine the dynamics of charged particle beams. It notes that these general subjects gave way to describing the motion of charged particles in beams. It investigates the notion of phase space, and the conservation of its density in Hamiltonian systems — the Louiville theorem. It introduces the concept of the design trajectory that allows nearby trajectories to be defined. It adds that the design trajectory also gives one the freedom to analyse the motion using distance along such a trajectory as the independent variable, instead of time.
J. B. Rosenzweig
- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198525547
- eISBN:
- 9780191711725
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525547.001.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This book presents beam physics using a unified approach, emphasizing basic concepts and analysis methods. While many existing resources in beams and accelerators are specialized to aid the ...
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This book presents beam physics using a unified approach, emphasizing basic concepts and analysis methods. While many existing resources in beams and accelerators are specialized to aid the professional practitioner, this text anticipates the needs of physics students. The central concepts underpinning the physics of accelerators, charged particle, and photon beams are built up from familiar, intertwining components, such as electromagnetism, relativity, and Hamiltonian dynamics. These components are woven into an illustrative set of examples that allow investigation of a variety of physical scenarios. With these tools, single particle dynamics in linear accelerators are discussed, with general methods that are naturally extended to circular accelerators. Beyond single particle dynamics, the proliferation of commonly used beam descriptions are surveyed and compared. These methods provide a powerful connection between the classical charged particle beams, and beams based on coherent waves — laser beams. Aspects of experimental techniques are introduced. Numerous exercises, and examples drawn from devices such as synchrotrons and free-electron lasers, are included to illustrate relevant physical principles.Less
This book presents beam physics using a unified approach, emphasizing basic concepts and analysis methods. While many existing resources in beams and accelerators are specialized to aid the professional practitioner, this text anticipates the needs of physics students. The central concepts underpinning the physics of accelerators, charged particle, and photon beams are built up from familiar, intertwining components, such as electromagnetism, relativity, and Hamiltonian dynamics. These components are woven into an illustrative set of examples that allow investigation of a variety of physical scenarios. With these tools, single particle dynamics in linear accelerators are discussed, with general methods that are naturally extended to circular accelerators. Beyond single particle dynamics, the proliferation of commonly used beam descriptions are surveyed and compared. These methods provide a powerful connection between the classical charged particle beams, and beams based on coherent waves — laser beams. Aspects of experimental techniques are introduced. Numerous exercises, and examples drawn from devices such as synchrotrons and free-electron lasers, are included to illustrate relevant physical principles.
Alfonso Sorrentino
- 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 ...
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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 some basic facts on Lagrangian and Hamiltonian dynamics (and on their mutual relation), which will be of fundamental importance in the discussion thereafter.Less
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 some basic facts on Lagrangian and Hamiltonian dynamics (and on their mutual relation), which will be of fundamental importance in the discussion thereafter.
Alfonso Sorrentino
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.001.0001
- Subject:
- Mathematics, Applied Mathematics
John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical ...
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John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry–Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, the book first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. The book achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. It then describes the whole theory and its subsequent developments and applications in their full generality.Less
John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry–Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, the book first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. The book achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. It then describes the whole theory and its subsequent developments and applications in their full generality.
J. B. Rosenzweig
- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198525547
- eISBN:
- 9780191711725
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525547.003.0002
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter is concerned with introducing a number of model problems, based on the relativistic motion of charged particles in static electric and magnetic field configurations. It discusses that ...
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This chapter is concerned with introducing a number of model problems, based on the relativistic motion of charged particles in static electric and magnetic field configurations. It discusses that these configurations include uniform dipole magnetic fields, uniform electric fields, quadrupole magnetic and electric fields, superpositions of uniform electric and magnetic fields, periodic magnetic dipole field or magnetic undulator. It explains that these model problems also allowed this chapter to introduce some rudimentary examples of analyses that are based on both relativistic and Hamiltonian formalisms. It adds that these analyses will help form the basis of more complex investigations of charged particle motion.Less
This chapter is concerned with introducing a number of model problems, based on the relativistic motion of charged particles in static electric and magnetic field configurations. It discusses that these configurations include uniform dipole magnetic fields, uniform electric fields, quadrupole magnetic and electric fields, superpositions of uniform electric and magnetic fields, periodic magnetic dipole field or magnetic undulator. It explains that these model problems also allowed this chapter to introduce some rudimentary examples of analyses that are based on both relativistic and Hamiltonian formalisms. It adds that these analyses will help form the basis of more complex investigations of charged particle motion.
Flavio Mercati
- Published in print:
- 2018
- Published Online:
- April 2018
- ISBN:
- 9780198789475
- eISBN:
- 9780191831294
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198789475.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Particle Physics / Astrophysics / Cosmology
The Hamiltonian formulation of relational particle dynamics unveils its equivalence with modern gauge theory, which admits exactly the same canonical formulation. Both are constrained Hamiltonian ...
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The Hamiltonian formulation of relational particle dynamics unveils its equivalence with modern gauge theory, which admits exactly the same canonical formulation. Both are constrained Hamiltonian systems with nonhonolomic constraints, for which Dirac’s analysis, made popular by his lectures, is necessary. Dirac’s analysis is briefly summarized in this chapter for readers unfamiliar with it. The Hamiltonian formulation of the kind of systems we’re interested in is nontrivial. In fact the standard formulation fails to be predictive, precisely because of the relational nature of our dynamics.Less
The Hamiltonian formulation of relational particle dynamics unveils its equivalence with modern gauge theory, which admits exactly the same canonical formulation. Both are constrained Hamiltonian systems with nonhonolomic constraints, for which Dirac’s analysis, made popular by his lectures, is necessary. Dirac’s analysis is briefly summarized in this chapter for readers unfamiliar with it. The Hamiltonian formulation of the kind of systems we’re interested in is nontrivial. In fact the standard formulation fails to be predictive, precisely because of the relational nature of our dynamics.
Vladimir Zeitlin
- Published in print:
- 2018
- Published Online:
- April 2018
- ISBN:
- 9780198804338
- eISBN:
- 9780191842627
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198804338.003.0013
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics
Main notions and ideas of wave (weak) turbulence theory are explained with the help of Hamiltonian approach to wave dynamics, and are applied to waves in RSW model. Derivation of kinetic equations ...
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Main notions and ideas of wave (weak) turbulence theory are explained with the help of Hamiltonian approach to wave dynamics, and are applied to waves in RSW model. Derivation of kinetic equations under random-phase approximation is explained. Short inertia–gravity waves on the f plane, short equatorial inertia–gravity waves, and Rossby waves on the beta plane are then considered along these lines. In all of these cases, approximate solutions of kinetic equation, annihilating the collision integral, can be obtained by scaling arguments, giving power-law energy spectra. The predictions of turbulence of inertia–gravity waves on the f plane are compared with numerical simulations initialised by ensembles of random waves. Energy spectra much steeper than theoretical are observed. Finite-size effects, which prevent energy transfer from large to short scales, provide a plausible explanation. Long waves thus evolve towards breaking and shock formation, yet the number of shocks is insufficient to produce shock turbulence.Less
Main notions and ideas of wave (weak) turbulence theory are explained with the help of Hamiltonian approach to wave dynamics, and are applied to waves in RSW model. Derivation of kinetic equations under random-phase approximation is explained. Short inertia–gravity waves on the f plane, short equatorial inertia–gravity waves, and Rossby waves on the beta plane are then considered along these lines. In all of these cases, approximate solutions of kinetic equation, annihilating the collision integral, can be obtained by scaling arguments, giving power-law energy spectra. The predictions of turbulence of inertia–gravity waves on the f plane are compared with numerical simulations initialised by ensembles of random waves. Energy spectra much steeper than theoretical are observed. Finite-size effects, which prevent energy transfer from large to short scales, provide a plausible explanation. Long waves thus evolve towards breaking and shock formation, yet the number of shocks is insufficient to produce shock turbulence.
Prasenjit Saha and Paul A. Taylor
- Published in print:
- 2018
- Published Online:
- July 2018
- ISBN:
- 9780198816461
- eISBN:
- 9780191858246
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198816461.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
Celestial mechanics abounds in interesting and counter-intuitive phenomena, such as descriptions of mass transfer between stars or optimal placements of satellites within the Solar System. ...
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Celestial mechanics abounds in interesting and counter-intuitive phenomena, such as descriptions of mass transfer between stars or optimal placements of satellites within the Solar System. Remarkably, many such features are already present in the restricted three-body problem, whose assumptions still allow for analytical understanding, and to which the second chapter is devoted. This ‘simplified’ system is discussed first in terms of forces (both gravitational and fictitious), and then using the Hamiltonian form. As well as traditional topics like stable and unstable Lagrange points and Roche lobes, a brief introduction to chaotic orbits is given. Additionally, readers are guided towards exploring on their own with numerical orbit integration.Less
Celestial mechanics abounds in interesting and counter-intuitive phenomena, such as descriptions of mass transfer between stars or optimal placements of satellites within the Solar System. Remarkably, many such features are already present in the restricted three-body problem, whose assumptions still allow for analytical understanding, and to which the second chapter is devoted. This ‘simplified’ system is discussed first in terms of forces (both gravitational and fictitious), and then using the Hamiltonian form. As well as traditional topics like stable and unstable Lagrange points and Roche lobes, a brief introduction to chaotic orbits is given. Additionally, readers are guided towards exploring on their own with numerical orbit integration.
