*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 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.

*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/9780198822370.003.0022
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic ...
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This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.Less

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

*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/9780198718062.003.0015
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter reviews the parametrization of absolute time and develop the concept of temporal reparametrizations, or changes in our clocks. Using the action integral and Dirac’s constraint mechanics, ...
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This chapter reviews the parametrization of absolute time and develop the concept of temporal reparametrizations, or changes in our clocks. Using the action integral and Dirac’s constraint mechanics, the chapter discusses the mathematics of temporal reparametrization and reparametrization form invariance. This latter is related to general covariance in general relativity. Working in an extended phase space that now includes time and its associated ‘momentum’, the chapter derives the result that the extended Hamiltonian is zero on the surface of constraints, that is, over the true or classical trajectory. The implications of this for quantum cosmology and the ‘problem of time’ discussed in Chapter 8 are examined and analysed in terms of contextual completeness.Less

This chapter reviews the parametrization of absolute time and develop the concept of temporal reparametrizations, or changes in our clocks. Using the action integral and Dirac’s constraint mechanics, the chapter discusses the mathematics of temporal reparametrization and reparametrization form invariance. This latter is related to general covariance in general relativity. Working in an extended phase space that now includes time and its associated ‘momentum’, the chapter derives the result that the extended Hamiltonian is zero on the surface of constraints, that is, over the true or classical trajectory. The implications of this for quantum cosmology and the ‘problem of time’ discussed in Chapter 8 are examined and analysed in terms of contextual completeness.