bluuebunny/crossref_metadata_2025_split huggingface dataset:
10 Small Vibrations about Equilibrium
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis chapter focuses on the one or more essentially stable equilibrium configurations of mechanical systems. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called normal modes. The Lagrangian theory of these small vibrations is presented here for the simple case of systems with a finite number of degrees of freedom. The chapter defines equilibrium by using the example of a marble. A marble placed at rest at the bottom of a spherical bowl will remain there forever. A marble placed at rest, and very carefully, on the top of a sphere will also remain there so long as no forces other than gravity act, as will a marble placed on a flat, level tabletop. These examples illustrate the three types of equilibrium point. The first is called stable; the second, unstable; and the third, conditional.
13 Lagrangian Mechanics with Time as a Coordinate
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis chapter presents an extended theory in which the traditional Lagrange equations and generalised energy theorem are combined into a single set of equations that restore the symmetry of the mathematical system. The traditional Lagrangian methods are analogous to the ‘coordinate parametric method’ in the calculus of variations, while the extended Lagrangian theory is analogous to the recommended ‘general parametric method’. The generalised energy theorem of traditional Lagrangian theory, which had been a separate equation analogous to the ‘second form’ of the Euler-Lagrange equations, gets restored to its proper place as just another of the extended Lagrange equations, which now form a complete set of equations appropriate to the problem.
Hamilton'S Principle and Noether's Theorem
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis chapter presents extended forms of Hamilton's Principle and the phase space Hamilton's principle based on the extended Lagrangian and Hamiltonian methods. Hamilton's Principle has already been treated in the context of traditional Lagrangian and Hamiltonian mechanics. Noether's theorem, a method for using symmetries of the extended Lagrangian to identify quantities that are conserved during the motion of the system, is also presented here. Noether's theorem is a powerful technique for discovering conserved quantities in complex Lagrangian systems. The chapter presents the basics of the method in the simple context of Lagrangian systems with a finite number of degrees of freedom. Both the traditional and the extended Hamilton's Principles are an application of the calculus of variations to mechanics. The traditional Hamilton's Principle uses the coordinate parametric method, while the extended Hamilton's Principle uses the general parametric method.
Relativity and Spacetime
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis chapter introduces the ideas of special relativity with the assumption that it is an established theory whose experimental efficacy is unquestioned. It attempts to provide a deeper understanding of the theory, and to determine what the relativistic effects may be telling us about the world and the nature of spacetime. The discussion here begins with the relativity principle of Newtonian physics, since it underlies all discussions of relativity. This relativity principle is commonly called Galilean relativity, although the same ideas are found in the Principia and in works before Galileo. Certainly, Galileo provided a detailed and poetic statement of it in a book of wide influence, the Dialog Concerning the Two Chief World Systems (Galilei, 1632).
19 Canonical Transformations
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis 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.
20 Generating Functions
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis chapter proposes methods for creating transformations that will automatically be canonical. Canonical transformations can be created by first choosing what are called generating functions. Using one of these generating functions in the formalism to be described will generate a transformation that will be canonical by construction. The generating functions can be quite general, leading to a wide selection of possible canonical transformations. Not only does every generating function lead to a canonical transformation, the converse is also true. Given any canonical transformation, a generating function can always be found that will generate it. The chapter begins by first relating canonical transformations to the existence of an intermediate function that will be called a protogenerating function.
21 Hamilton-Jacobi Therory
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis chapter focuses on the Hamilton-Jacobi theory, the apotheosis of Lagrangian and Hamiltonian mechanics. Action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria, and are the solutions of a non-linear, first-order partial differential equation called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. An entire class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This connection is discussed in detail in the chapter, together with the Bohm hidden variable model and Feynman path integral method that are derived from it.
22 Angle‐Action Variables
Authors: Johns Oliver Davis 
Published: December 2013

AbstractThis chapter introduces the angle-action method and provides several examples of its use, including its essential role in the ‘old’ quantum theory, a precursor to the Schroedinger wave equation that sought to introduce quantum ideas into a classical model. Angle-action methods are also used in what is called canonical perturbation theory, the study of systems that deviate only slightly from cyclic ones. That vast subject is best studied using the language of differential geometry. Study of the present chapter will give the reader a firm grasp of the angle-action foundation upon which canonical perturbation theory is constructed. Certain specialised systems of importance in astronomy and quantum theory can be treated by a canonical transformation to angle-action variables.
Riemann Surfaces
Authors: Donaldson Simon 
Published: December 2013

Abstract
The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved.
The Uniformisation Theorem
Authors: Donaldson Simon 
Published: December 2013

Abstract
This chapter proves the following theorem. Theorem 12: Let X be a connected, simply connected, non-compact Riemann surface. Then X is equivalent to either C or the upper half-plane H. The proof presented here follows the same general pattern as one already given to classify compact simply connected Riemann surfaces, but the non-compactness will require some extra steps.
Contrasts in Riemann surface theory
Authors: Donaldson Simon 
Published: December 2013

Abstract
Section 4.2.3 showed how to associate a compact connected Riemann surface to any irreducible polynomial P(z,w) in two variables. This chapter first establishes the converse — any compact connected Riemann surface arises from a polynomial P. It also explains how the whole theory can be cast in an algebraic form, involving fields of transcendence degree 1. The chapter then analyzes hyperbolic surfaces, covering models of the hyperbolic plane; self-isometries; geodesics; the Gauss-Bonnet Theorem; right-angled hexagons; and closed geodesics.
Waves, complex numbers and Fourier transforms
Authors: Sivia D.S. 
Published: December 2013

Abstract
This chapter provides a tutorial introduction to the basic physical concepts, and the associated analytical tools, needed for an understanding of wave phenomena. An everyday description of a wave would be a ‘wiggle’, or something that goes up-and-down as you move forward. The progression of the fluctuations could refer to changes in ‘height’ with respect to position at a fixed time, or with respect to time at a fixed position. Several examples of geometrical waves are presented here; they are unusual in that they have points where there are abrupt changes in the value of the function or its gradient. What they have in common with the more familiar sinusoidal variation is a regularly repeating pattern.
Small-angle scattering and the big picture
Authors: Sivia D. S. 
Published: December 2013

Abstract
This chapter focuses on small-angle scattering, which is used to obtain information regarding the size, shape, and assemblies of large molecules and aggregates. Most small-angle scattering experiments are carried out on dilute solutions, so that the sample consists of isolated and isotropically oriented examples of the object of interest located randomly within a sea of solvent. The chapter also discusses the Guinier relationship, which posits that the scattered signal is insensitive to the shape of the solute particles at the smallest Qs; only their overall size, in an rms sense, and the content of the sample, matter. As with all Fourier data, information about structural detail is found at higher Qs. Even then, distinction can be difficult since differences are blurred out by orientational averaging.
Energy exchange and dynamical information
Authors: Sivia D. S. 
Published: December 2013

Abstract
This chapter discusses the concepts of X-ray and neutron scattering, considering the general case where there is an exchange of energy as well as momentum. The work of Brockhouse is followed here, wherein inelastic experiments are traditionally carried out with a triple-axis spectrometer. It differs from an elastic, or double-axis, setup through the addition of an analyser stage, since the final and incident energies are no longer assumed to be the same. The analyser is simply a monochromator that intercepts the scattered beam. Both selectively reflect particles of a given wavelength based on the Bragg condition satisfied by suitably oriented crystals. Although a triple-axis spectrometer is very versatile, each monochromation stage takes a heavy toll on the particle flux. One of the two can be avoided naturally at a spallation neutron source by using the time-of-flight technique.
Examples of inelastic scattering
Authors: Sivia D. S. 
Published: December 2013

Abstract
This chapter, which considers a number of examples of inelastic scattering, aims to gain an appreciation of the types of dynamical behaviour that can be probed with X-rays and neutrons, without being comprehensive or too theoretical. The earliest example of inelastic scattering at the atomic level dates back to 1922, with the work of Compton on the scattering of X-rays by a thin foil of graphite. He found that the signal at any given scattering angle contained two wavelengths: one was equal to that of the incident monochromatic beam, as expected from classical radiation theory, but the other was longer. Compton explained this component of the scattering by considering it to be the result of a billiard ball-type of collision between a photon and a stationary electron.
Exploring Quantum Mechanics
Authors: Galitski Victor , Karnakov Boris , Kogan Vladimir 
Published: December 2013

Abstract
A series of seminal technological revolutions has led to a new generation of electronic devices miniaturized to such tiny scales that the strange laws of quantum physics come into play. There is no doubt that, unlike scientists and engineers of the past, technology leaders of the future will have to rely on quantum mechanics in their everyday work. This makes teaching and learning the subject of paramount importance for further progress. Mastering quantum physics is a very non-trivial task and its deep understanding can only be achieved through working out real-life problems and examples. It is notoriously difficult to come up with new quantum-mechanical problems that would be solvable with a pencil and paper, and within a finite amount of time. This book presents some 700+ original problems in quantum mechanics together with detailed solutions, covering nearly 1,000 pages on all aspects of quantum science.
One-dimensional motion
Authors: Galitski Victor , Karnakov Boris , Kogan Vladimir , Galitski, Jr. Victor 
Published: December 2013

Abstract
This chapter analyzes problems that deal with stationary states in the discrete spectrum; the Schrödinger equation in momentum space (the Green function and integral form of the Schrödinger equation); the continuous spectrum (reflection from and transmission through potential barriers); and systems with several degrees of freedom (particle in a periodic potential).
Perturbation theory; Variational method; Sudden and adiabatic theory
Authors: Galitski Victor , Karnakov Boris , Kogan Vladimir , Galitski, Jr. Victor 
Published: December 2013

Abstract
This chapter analyzes problems that deal with stationary perturbation theory (discrete spectrum); variational method; stationary perturbation theory (continuous spectrum); non-stationary perturbation theory (transitions in continuous spectrum); sudden perturbations; and adiabatic approximation.
Prologue
Authors: Moore Cristopher , Mertens Stephan 
Published: December 2013

AbstractThis prologue begins by considering three examples to demonstrate that in order to solve different problems, fundamentally different kinds of search and different types of proof are required. The first example deals with the nature of computation with a walk through the eighteenth-century town of Königsberg (now Kaliningrad), which had seven bridges connecting the two banks of the river Pregel with two islands. A popular puzzle of the time was whether it is possible to walk through the city by crossing each bridge only once. This puzzle was solved by Leonhard Euler in 1736 in the form of a theorem which states that: A connected graph contains an Eulerian cycle if and only if every vertex has even degree. If exactly two vertices have odd degree, it contains an Eulerian path but not an Eulerian cycle. The second example deals with Hamiltonian paths or cycles, while the third involves factoring integers and chess problems. This book explores how to solve problems as efficiently as possible — and how, and why, some problems are extremely hard.
Insights and Algorithms
Authors: Moore Cristopher , Mertens Stephan 
Published: December 2013

AbstractThere are many forms of mathematical insights, but only a few major strategies can be used to construct polynomial-time algorithms. These include divide and conquer, dynamic programming, greedy algorithms, duality, and reductions. This chapter explores these strategies and considers how a problem can be broken into subproblems that are small enough, and few enough, to solve quickly. It presents examples that demonstrate how to sort a pack of cards, align genomes, find short paths, hear the music of the spheres, route the flow of traffic, typeset beautiful books, build efficient networks, or run a dating service. The chapter begins with a classic example of recursion: the Towers of Hanoi, introduced by the mathematician Edouard Lucas. It then looks at several important problems where a divide-and-conquer strategy works and explains how so many other problems can be expressed in terms of reachability and shortest path.
Who is the Hardest One of All? NP-Completeness
Authors: Moore Cristopher , Mertens Stephan 
Published: December 2013

AbstractThere are problems that cannot be solved, including 3-SAT, graph coloring, and Hamiltonian path. Each of these problems has the remarkable ability to express all the others, or any other problem in NP. This remarkable property is known as NP-completeness. This chapter examines the concept of NP-completeness and considers examples of NP-complete problems, from coloring maps to evaluating integrals. First, it translates the witness-checking programs of WITNESS EXISTENCE into simpler mathematical objects: Boolean circuits and formulas. It then reduces WITNESS EXISTENCE to 3-SAT, one of the most basic constraint satisfaction problems. It also demonstrates that reductions work by thinking about maps from solutions to solutions, explores some additional problems whose NP-completeness is somewhat surprising, and shows that NP-completeness exists even in mathematics. The chapter concludes by proving that Hamiltonian path is NP-complete.
Optimization and Approximation
Authors: Moore Cristopher , Mertens Stephan 
Published: December 2013

AbstractThis chapter focuses on the relationships between decision problems and their optimisation versions. It shows that, for most problems, the optimal solution can be realised in polynomial time if and only if we can tell whether a solution with a given quality exists. It then explores approximation algorithms for the traveling salesman problem and considers some large families of optimisation problems that can be solved in polynomial time, including linear programming and semidefinite programming. It demonstrates that the duality between MAX FLOW and MIN CUT is no accident — that linear programming problems come in pairs. In addition, the chapter looks at integer linear programming, which can be solved in polynomial time before concluding with a discussion of optimisation problems from a practical point of view.
Randomized Algorithms
Authors: Moore Cristopher , Mertens Stephan 
Published: December 2013

AbstractCertain situations require a random rather than a deterministic strategy. With a random strategy, the choices are unpredictable and the adversary may be kept off balance. This chapter focuses on the variety and power of randomised algorithms. More specifically, it considers algorithms that find the smallest cut in a graph by combining random vertices until only two remain, or play games by searching the tree of possible moves in random order. It also explains how to check whether the software on a space probe is uncorrupted, how to turn a puzzle with many solutions into one with a unique solution, and how to determine whether two functions are equal. It describes tools such as random hash functions and polynomial identity testing, analyzes the nature of the primes, and looks at a series of randomised algorithms for primality based on different number-theoretic ideas. The chapter concludes by discussing several complexity classes consisting of problems that can be solved using various kinds of randomized algorithms in polynomial time.
Quantum Computation
Authors: Moore Cristopher , Mertens Stephan 
Published: December 2013

AbstractAccording to the Church-Turing Thesis, a Turing machine, and thus the programming languages and computers in existence, can simulate any reasonable computing device. However, there are physical devices that are exponentially more efficient than a Turing machine, and hence would prove exponentially difficult to simulate using such a machine. This chapter explores the concept of quantum computers and their potential not only to simulate quantum physics exponentially faster than classical computers but also to solve other problems exponentially faster. It considers algorithms that break cryptosystems, find needles in haystacks, and predict who can win a game. It first describes particles, waves, and amplitudes in quantum computation before turning to states and operators. It then examines how interference lets a quantum computer learn some things about a function, how quantum computers can solve factoring in polynomial time, the link between factoring and modern cryptography, graph isomorphism and the hidden subgroup problem, and quantum walks and scattering.
Introduction
Authors: Kirchman David L. 
Published: December 2013

Abstract
This chapter introduces the field of microbial ecology and some terms used in the rest of the book. Microbial ecology, which is the study of microbes in natural environments, is important for several reasons. Although most are beneficial, some microbes cause diseases of higher plants and animals in aquatic environments and on land. Microbes are also important because they are directly or indirectly responsible for the food we eat. They degrade pesticides and other pollutants contaminating natural environments. Finally, microbes are important in another ‘pollution’ problem: the increase in greenhouse gases such as carbon dioxide and methane in the atmosphere. Because microbes are crucial for many biogeochemical processes, the field of microbial ecology is crucial for understanding the effect of greenhouse gases on the biosphere and for predicting the impact of climate change on aquatic and terrestrial ecosystems. Even if the problem of climate change was solved, microbes would be fascinating to study because of the weird and wonderful things they do. The chapter ends by pointing out the difficulties in isolating and cultivating microbes in the lab. In many environments, 〈 1 per cent of all bacteria and probably other microbes can be grown in the lab. The cultivation problem has many ramifications for identifying especially viruses, bacteria, and archaea in natural environments and for connecting up taxonomic information with biogeochemical processes.