Search Results

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.

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.

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.

The chapter begins with a phenomenological treatment of the observed atmospheric circulation. It then goes on to discuss how the barotropic model arises as a so-calledbalanced model of the slow, vorticity-driven dynamics, from the more general shallowwater model which also admits inertia-gravity waves. This is important because large-scale atmospheric turbulence exhibits aspects of both balanced and unbalanced dynamics. Because of the first-order importance of zonal flows in the atmospheric general circulation, the large-scale turbulence is highly inhomogeneous, and is shaped by the nature of the interaction between zonal flows and Rossby waves described eloquently by Michael McIntyre as a wave-turbulence jigsaw puzzle. This motivates a review of the barotropic theory of wave, mean-flow interaction, which is underpinned by the Hamiltonian structure of geophysical fluid dynamics.

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.

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.

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.