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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, δ_I), where I is an integrable complex structure tensor and δ_I is a holomorphic Poisson tensor. (I, δ_I) 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, δ_I) which does not imply the holomorphic equivalence of the underlying complex structures.

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.

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.

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.

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.

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 in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.

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 in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.

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.

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.

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.

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.

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, and efforts to highlight the links between the two approaches have been made. A wide range of illustrative examples with a particular focus on non-holonomic systems are studied, including Chaplygin’s sleigh, rolling balls, and rolling discs. Equilibria and stability, and the connection between timereversal symmetry and dissipation, are also briefly discussed.

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.

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.

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.

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.

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 author’s perception of what potential users of this text were exposed to in their earlier studies. Therefore, only brief overview is given of some subjects while somewhat more comprehensive discussion is given of others. In neither case can the coverage provided substitute for the actual learning of these subjects that are covered in detail by many textbooks. A random variable is an observable whose repeated determination yields a series of numerical values (“realizations” of the random variable) that vary from trial to trial in a way characteristic of the observable. The outcomes of tossing a coin or throwing a die are familiar examples of discrete random variables. The position of a dust particle in air and the lifetime of a light bulb are continuous random variables. Discrete random variables are characterized by probability distributions; Pn denotes the probability that a realization of the given random variable is n. Continuous random variables are associated with probability density functions P(x): P(x1)dx denotes the probability that the realization of the variable x will be in the interval x1 . . . x1+dx.

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 the distribution functions are known. Throughout, we shall make use of the ‘rigid molecule’ and classical approximations. In the rigid molecule approximation the system intermolecular potential energy u(rNωN ) depends only on the positions of the centres of mass rN ≡ r1 . . . rN for the N molecules and on their molecular orientations ωN ≡ ω1 . . . ωN; any dependence on vibrational or internal rotational coordinates is neglected. In the classical approximation the translational and rotational motions of the molecules are assumed to be classical. These assumptions should be quite realistic for many fluids composed of simple molecules, e.g. N2 , CO, CO2 , SO2 CF4 , etc. They are discussed in detail in §§ 1.2.1 and 1.2.2; quantum corrections to the partition function are discussed in §§ 1.2.2 and 6.9, and in Appendix 3D. In considering fluids in equilibrium we can distinguish three principal cases: (a) isotropic, homogeneous fluids (e.g. liquid or compressed gas states of N2 , O2 , etc. in the absence of an external field), (b) anisotropic, homogeneous fluids (e.g. a polyatomic fluid in the presence of a uniform electric field, nematic liquid crystals), and (c) inhomogeneous fluids (e.g. the interfacial region). These fluid states have been listed in order of increasing complexity; thus, more independent variables are involved in cases (b) and (c), and consequently the evaluation of the necessary distribution functions is more difficult. For molecular fluids it is convenient to introduce several types of distribution functions, correlation functions, and related quantities: (a) The angular pair correlation function g(r1r2 ω1 ω2). This gives complete information about the pair of molecules, and arises in expressions for the equilibrium properties for a general potential.