Marco Gualtieri
- 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 ...
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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 bracket on its sheaf of regular functions. The structure is determined, on a real smooth manifold M, by the choice of a pair (I, δI), where I is an integrable complex structure tensor and δI is a holomorphic Poisson tensor. (I, δI) is viewed as a generalized complex structure, in the sense of Hitchin. In viewing it in this way, the chapter provides a new notion of equivalence between the pairs (I, δI) which does not imply the holomorphic equivalence of the underlying complex structures.Less
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 bracket on its sheaf of regular functions. The structure is determined, on a real smooth manifold M, by the choice of a pair (I, δI), where I is an integrable complex structure tensor and δI is a holomorphic Poisson tensor. (I, δI) is viewed as a generalized complex structure, in the sense of Hitchin. In viewing it in this way, the chapter provides a new notion of equivalence between the pairs (I, δI) which does not imply the holomorphic equivalence of the underlying complex structures.
Oliver Johns
- 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 ...
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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 coordinate or momentum is allowed to be a function of all of the previous phase-space coordinates, including the previous canonical momenta. Thus the new position and time variables may depend, through their dependence on the momenta, on the old velocities as well as the old positions. The Lagrange equations will not in general be form invariant under such transformations. Canonical transformations are the most general phase-space transformations that preserve the extended Hamilton equations. There are several equivalent definitions of canonical transformations, three of which are the Poisson bracket condition, the direct condition, and the Lagrange bracket condition. The definition of canonical transformation includes the Lorentz transformation of special relativity. Also discussed are symplectic coordinates, form invariance of Poisson brackets, and form invariance of the Hamilton equations.Less
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 coordinate or momentum is allowed to be a function of all of the previous phase-space coordinates, including the previous canonical momenta. Thus the new position and time variables may depend, through their dependence on the momenta, on the old velocities as well as the old positions. The Lagrange equations will not in general be form invariant under such transformations. Canonical transformations are the most general phase-space transformations that preserve the extended Hamilton equations. There are several equivalent definitions of canonical transformations, three of which are the Poisson bracket condition, the direct condition, and the Lagrange bracket condition. The definition of canonical transformation includes the Lorentz transformation of special relativity. Also discussed are symplectic coordinates, form invariance of Poisson brackets, and form invariance of the Hamilton equations.
Oliver Johns
- 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 ...
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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 Lagrange equations and the generalised energy theorem into one set of equations. In the present chapter, the same is done with the traditional Hamilton equations. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, are combined into a single set of extended Hamilton equations in which time is treated as a coordinate. The chapter also discusses the extended phase space, dependency relation, shift from traditional to extended Hamiltonian mechanics, equivalence to traditional Hamilton equations, example of extended Hamilton equations, equivalent extended Hamiltonians, alternate Hamiltonians, alternate traditional Hamiltonians, Dirac’s theory of phase-space constraints, Poisson brackets with time as a coordinate, and Poisson brackets and quantum commutators.Less
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 Lagrange equations and the generalised energy theorem into one set of equations. In the present chapter, the same is done with the traditional Hamilton equations. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, are combined into a single set of extended Hamilton equations in which time is treated as a coordinate. The chapter also discusses the extended phase space, dependency relation, shift from traditional to extended Hamiltonian mechanics, equivalence to traditional Hamilton equations, example of extended Hamilton equations, equivalent extended Hamiltonians, alternate Hamiltonians, alternate traditional Hamiltonians, Dirac’s theory of phase-space constraints, Poisson brackets with time as a coordinate, and Poisson brackets and quantum commutators.
Oliver Davis Johns
- 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 ...
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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 transformations; three of them are presented here: the Poisson bracket condition, the direct condition, and the Lagrange bracket condition. Each of these three conditions has two forms, a long one that is written out in terms of partial derivatives and a symplectic one consisting of a single matrix equation. The chapter begins with the long form of the Lagrange bracket condition, and after introducing some necessary notation, derives the long and symplectic forms of all three. The definition of canonical transformation includes the Lorentz transformation of special relativity, since we are now operating in an extended phase space.Less
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 transformations; three of them are presented here: the Poisson bracket condition, the direct condition, and the Lagrange bracket condition. Each of these three conditions has two forms, a long one that is written out in terms of partial derivatives and a symplectic one consisting of a single matrix equation. The chapter begins with the long form of the Lagrange bracket condition, and after introducing some necessary notation, derives the long and symplectic forms of all three. The definition of canonical transformation includes the Lorentz transformation of special relativity, since we are now operating in an extended phase space.
Oliver Johns
- 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 ...
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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 formulation is a much better base from which to build more advanced methods. The Hamilton equations have an elegant symmetry that the Lagrange equations lack. Another answer, not directly related to classical mechanics, is that the Hamiltonian function is used to write the Schroedinger equation of quantum mechanics. The differences between the Lagrange and Hamilton equations result mainly from the different variable sets in which they act. This chapter deals with phase space, Hamilton equations, example of the Hamilton equations, non-potential and constraint forces, reduced Hamiltonian, Poisson brackets, Schroedinger equation, and Ehrenfest theorem.Less
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 formulation is a much better base from which to build more advanced methods. The Hamilton equations have an elegant symmetry that the Lagrange equations lack. Another answer, not directly related to classical mechanics, is that the Hamiltonian function is used to write the Schroedinger equation of quantum mechanics. The differences between the Lagrange and Hamilton equations result mainly from the different variable sets in which they act. This chapter deals with phase space, Hamilton equations, example of the Hamilton equations, non-potential and constraint forces, reduced Hamiltonian, Poisson brackets, Schroedinger equation, and Ehrenfest theorem.
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 ...
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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 beauty of elliptic curves, which will also appear later in rigid body mechanics. Geodesics in Riemannian geometry are presented as an example of a Hamiltonian system. Conversely, the path of a non-relativistic particle is a geodesic in a metric that depends on the potential. Orbits around a black hole are found. Hamilton-Jacobi theory is discussed, showing a path towards quantum mechanics and a connection to the eikonal of optics. The three body problem is studied in detail, including small orbits around the Lagrange points. The dynamics of a charged particle in a magnetic field, especially a magnetic monopole, is studied in the Hamiltonian formalism. Spin is shown to be a classical phenomenon. Symplectic integrators that allow numerical solutions of mechanical systems are derived. A simplified version of Feigenbaum's theory of period doubling introduces chaos. Following a classification of Mobius transformations, this book studies chaos on the complex plane: Julia sets, Fatou sets, and the Mandelblot are explained. Newton's method for solution of non-linear equations is viewed as a dynamical system, allowing a novel approach to the reduction of matrices to canonical form. This is used as a stepping stone to the KAM theory of maps of a circle to itself, unravelling a connection to the Diophantine problem of number theory. KAM theory of the solution of the Hamilton-Jacobi equation using Newton's iteration concludes the book.Less
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 beauty of elliptic curves, which will also appear later in rigid body mechanics. Geodesics in Riemannian geometry are presented as an example of a Hamiltonian system. Conversely, the path of a non-relativistic particle is a geodesic in a metric that depends on the potential. Orbits around a black hole are found. Hamilton-Jacobi theory is discussed, showing a path towards quantum mechanics and a connection to the eikonal of optics. The three body problem is studied in detail, including small orbits around the Lagrange points. The dynamics of a charged particle in a magnetic field, especially a magnetic monopole, is studied in the Hamiltonian formalism. Spin is shown to be a classical phenomenon. Symplectic integrators that allow numerical solutions of mechanical systems are derived. A simplified version of Feigenbaum's theory of period doubling introduces chaos. Following a classification of Mobius transformations, this book studies chaos on the complex plane: Julia sets, Fatou sets, and the Mandelblot are explained. Newton's method for solution of non-linear equations is viewed as a dynamical system, allowing a novel approach to the reduction of matrices to canonical form. This is used as a stepping stone to the KAM theory of maps of a circle to itself, unravelling a connection to the Diophantine problem of number theory. KAM theory of the solution of the Hamilton-Jacobi equation using Newton's iteration concludes the book.
S. G. Rajeev
- 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 ...
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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 lead to conservation laws. A glimpse of the star product of quantum mechanics is given: the classical Poisson bracket is its infinitesimal approximation. In an exercise, the quantum harmonic oscillator is solved in the star product formalism.Less
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 lead to conservation laws. A glimpse of the star product of quantum mechanics is given: the classical Poisson bracket is its infinitesimal approximation. In an exercise, the quantum harmonic oscillator is solved in the star product formalism.
Rodolfo Gambini and Jorge Pullin
- 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. ...
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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. It introduces the application of Hamiltonian methods to field theories and the notion of functional derivatives. The chapter presents Maxwell theory as an example and discusses Gauss' law as a constraint generator of gauge transformations. It concludes with a discussion of totally constrained systems, corresponding to the situation when the total Hamiltonian is a pure linear combination of constraints, and how to recover physics in systems without time.Less
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. It introduces the application of Hamiltonian methods to field theories and the notion of functional derivatives. The chapter presents Maxwell theory as an example and discusses Gauss' law as a constraint generator of gauge transformations. It concludes with a discussion of totally constrained systems, corresponding to the situation when the total Hamiltonian is a pure linear combination of constraints, and how to recover physics in systems without time.
Peter Mann
- 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 ...
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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 developed by the French mathematician Poisson in the late nineteenth century and it is a reformulation, or at least a tidying up, of Hamilton’s equations into one neat package. The Poisson bracket of a quantity with the Hamiltonian describes the time evolution of that quantity as one moves along a curve in phase space. The Lie algebra structure of symmetries in mechanics is highlighted using this formulation. The classical propagator is derived using the Poisson bracket.Less
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 developed by the French mathematician Poisson in the late nineteenth century and it is a reformulation, or at least a tidying up, of Hamilton’s equations into one neat package. The Poisson bracket of a quantity with the Hamiltonian describes the time evolution of that quantity as one moves along a curve in phase space. The Lie algebra structure of symmetries in mechanics is highlighted using this formulation. The classical propagator is derived using the Poisson bracket.
Tom Lancaster and Stephen J. Blundell
- 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 ...
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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 mechanics and Poisson brackets, the attention is shifted from the Lagrangian to the Lagrangian density. The chapter uses the electromagnetic field as a first example of this approach.Less
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 mechanics and Poisson brackets, the attention is shifted from the Lagrangian to the Lagrangian density. The chapter uses the electromagnetic field as a first example of this approach.
George Jaroszkiewicz
- 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 ...
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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 to discuss Aristotelian and Galilean–Newtonian space-times. It explains how Newton’s first law of motion is required to correlate successive copies of space. It discusses the differences between the Newtonian paradigm of motion and the use of Lagrangians in the Calculus of Variations that gives the Euler–Lagrange equations of motion in advanced mechanics. The chapter concludes with a discussion of phase space, Hamiltonians, Poisson brackets, and the role of the Hamiltonian as a generator of translation in time.Less
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 to discuss Aristotelian and Galilean–Newtonian space-times. It explains how Newton’s first law of motion is required to correlate successive copies of space. It discusses the differences between the Newtonian paradigm of motion and the use of Lagrangians in the Calculus of Variations that gives the Euler–Lagrange equations of motion in advanced mechanics. The chapter concludes with a discussion of phase space, Hamiltonians, Poisson brackets, and the role of the Hamiltonian as a generator of translation in time.
Jennifer Coopersmith
- 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 ...
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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 mathematical prodigy William Rowan Hamilton, inspired by Descartes, described the geometry of light rays algebraically. He extended this method to the motion of particles—his optico-mechanical theory—showing that particles exhibited wave features, but now ‘action’ was minimized instead of time. When a dual system of generalized coordinates (q and p) was used, and when T in the Lagrangian was normalized to a special form, a new energy function emerged—the Hamiltonian, H. The form of the Hamiltonian determined the evolution of the given mechanical system. The ‘canonical’ coordinates p and q occurred in an imaginary (virtual) phase space, and mapped out the range of mechanical possibilities. The links to Poisson brackets and Schrodinger’s wave mechanics are mentioned.Less
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 mathematical prodigy William Rowan Hamilton, inspired by Descartes, described the geometry of light rays algebraically. He extended this method to the motion of particles—his optico-mechanical theory—showing that particles exhibited wave features, but now ‘action’ was minimized instead of time. When a dual system of generalized coordinates (q and p) was used, and when T in the Lagrangian was normalized to a special form, a new energy function emerged—the Hamiltonian, H. The form of the Hamiltonian determined the evolution of the given mechanical system. The ‘canonical’ coordinates p and q occurred in an imaginary (virtual) phase space, and mapped out the range of mechanical possibilities. The links to Poisson brackets and Schrodinger’s wave mechanics are mentioned.
Nathalie Deruelle and Jean-Philippe Uzan
- 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 ...
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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 governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.Less
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 governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.
Peter Mann
- 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 ...
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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 equal footing but, in the canonical formalism, time is treated as being special and therefore, by definition, it is not covariant. Consequently, most field theoretic models are built on Lagrangian formulations. A covariant canonical formalism is the subject of the de Donder–Weyl formalism, which is briefly discussed as a covariant Hamiltonian field theory. In addition, the chapter examines the case of a generalised Poisson bracket in the continuous form for two local smooth functionals of phase space.Less
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 equal footing but, in the canonical formalism, time is treated as being special and therefore, by definition, it is not covariant. Consequently, most field theoretic models are built on Lagrangian formulations. A covariant canonical formalism is the subject of the de Donder–Weyl formalism, which is briefly discussed as a covariant Hamiltonian field theory. In addition, the chapter examines the case of a generalised Poisson bracket in the continuous form for two local smooth functionals of phase space.
