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1
NATO Advanced Study Institute on Sixty-two Years of Uncertainty: Historical, Philosophical, and Physical Inquiries into the Foundations of Quantum Mechanics (1989 Erice, Italy). Sixty-two years of uncertainty: Historical, philosophical, and physical inquiries into the foundations of quantum mechanics. New York: Plenum Press, 1990.
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2
de, Broglie Louis. Heisenberg's uncertainties and the probabilistic interpretation of wave mechanics: With critical notes of the author. Dordrecht: Kluwer Academic Publishers, 1990.
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3
Das Prinzip der kleinsten Wirkung und die Kraftkonzeptionen der rationalen Mechanik: Eine Untersuchung zur Grundlegungsproblematik by Leonhard Euler, Pierre Louis Moreau de Maupertius und Joseph Louis Lagrange. Stuttgart: F. Steiner, 1989.
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4
Drexel Symposium on Quantum Nonintegrability (4th 1994 Philadelphia, Pa.). Quantum classical correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability, Drexel University, Philadelphia, USA, September 8-11, 1994. Cambridge, MA: International Press, 1997.
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5
Buzon, Frédéric de. Descartes et les "Principia" II: Corps et mouvement. Paris: Presses universitaires de France, 1994.
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6
Coopersmith, Jennifer. Lagrangian Mechanics. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0006.
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It is demonstrated how d’Alembert’s Principle can be used as the basis for a more general mechanics – Lagrangian Mechanics. How this leads to Hamilton’s Principle (the Principle of Least Action) is shown mathematically and in words. It is further explained why Lagrangian Mechanics is so general, why forces of constraint may be ignored, and how external conditions lead to “curved space.” Also, it is explained why the Lagrangian, L, has the form L = T − V (where T is the kinetic energy and V is the potential energy), and why T is in “quadratic form” (T = 1/2mv2). It is shown how Noether’s Theorem leads to a more fundamental definition of energy and links the conservation of energy to the homogeneity of time. The ingenious Lagrange multipliers are explained, and also generalized forces and generalized coordinates.
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Deruelle, Nathalie, and Jean-Philippe Uzan. Lagrangian mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0008.
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This chapter shows how the Newtonian law of motion of a particle subject to a gradient force derived from a ‘potential energy’ can always be obtained from an extremal principle, or ‘principle of least action’. According to Newton’s first law, the trajectory representing the motion of a free particle between two points p1 and p2 is a straight line. In other words, out of all the possible paths between p1 and p2, the trajectory effectively followed by a free particle is the one that minimizes the length. However, even though the use of the principle of extremal length of the paths between two points gives the straight line joining the points, this does not mean that the straight-line path is traced with constant velocity in an inertial frame. Moreover, the trajectory describing the motion of a particle subject to a force is not uniform and rectilinear.
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Mann, Peter. Canonical & Gauge Transformations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0018.
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In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions to the Hamilton–Jacobi equations are discussed. The canonical action is examined in a fashion similar to that used for analysis in previous chapters. The Hamilton–Jacobi equation is then shown to parallel the eikonal equation of wave mechanics. The chapter discusses Hamilton’s principal function, the time-independent Hamilton–Jacobi equation, Hamilton’s characteristic function, the rectification theorem, the Maupertius action principle and the Hamilton–Jacobi variational problem. The chapter also discusses integral surfaces, complete integral hypersurfaces, completely separable solutions, the Arnold–Liouville integrability theorem, general integrals, the Cauchy problem and de Broglie–Bohm mechanics. In addition, an interdisciplinary example of medical imaging is detailed.
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Bitbol, Michel, Babette Babich, and Patrick Aidan Heelan. Observable: Heisenberg's Philosophy of Quantum Mechanics. Lang AG International Academic Publishers, Peter, 2015.
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10
Mercati, Flavio. Best Matching: Technical Details. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198789475.003.0005.
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The best matching procedure described in Chapter 4 is equivalent to the introduction of a principal fibre bundle in configuration space. Essentially one introduces a one-dimensional gauge connection on the time axis, which is a representation of the Euclidean group of rotations and translations (or, possibly, the similarity group which includes dilatations). To accommodate temporal relationalism, the variational principle needs to be invariant under reparametrizations. The simplest way to realize this in point–particle mechanics is to use Jacobi’s reformulation of Mapertuis’ principle. The chapter concludes with the relational reformulation of the Newtonian N-body problem (and its scale-invariant variant).
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Coopersmith, Jennifer. Hamiltonian Mechanics. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0007.
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Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.
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Tiwari, Sandip. Information mechanics. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198759874.003.0001.
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Information is physical, so its manipulation through devices is subject to its own mechanics: the science and engineering of behavioral description, which is intermingled with classical, quantum and statistical mechanics principles. This chapter is a unification of these principles and physical laws with their implications for nanoscale. Ideas of state machines, Church-Turing thesis and its embodiment in various state machines, probabilities, Bayesian principles and entropy in its various forms (Shannon, Boltzmann, von Neumann, algorithmic) with an eye on the principle of maximum entropy as an information manipulation tool. Notions of conservation and non-conservation are applied to example circuit forms folding in adiabatic, isothermal, reversible and irreversible processes. This brings out implications of fluctuation and transitions, the interplay of errors and stability and the energy cost of determinism. It concludes discussing networks as tools to understand information flow and decision making and with an introduction to entanglement in quantum computing.
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Sixty-Two Years of Uncertainty: Historical Philosophical, and Physical Enquiries into the Foundations of Quantum Mechanics (NATO Science Series: B:). Springer, 1990.
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14
Horing, Norman J. Morgenstern. Schwinger Action Principle and Variational Calculus. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0004.
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Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
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Levin, Frank S. Macroscopic Manifestations of Quantum Mechanics. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198808275.003.0013.
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Some possibly unexpected macroscopic manifestations of quantum mechanics are described in Chapter 12. The first is a laser, a device both man-made and one that relies on phase effects to achieve its potent beam. How this is done is illustrated by a diagram. The next is an estimate of the maximum height of a mountain, whose result was originally shown to rely on quantum mechanics. That result, approximately 30 km, is followed by showing that white dwarf and neutron stars are each gigantic manifestations of the Pauli Exclusion Principle, the first mainly consisting of carbon nuclei and electrons, the second mainly of neutrons. In each case, the primary constituent is a fermion, whose quantum behavior is governed by the Exclusion Principle. Along the way to showing this is a review of stellar evolution and energy sources. The final example is the first quantum machine, which is barely macroscopic.
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Zirnbauer, Martin R. Symmetry classes. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.3.
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This article examines the notion of ‘symmetry class’, which expresses the relevance of symmetries as an organizational principle. In his 1962 paper The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Dyson introduced the prime classification of random matrix ensembles based on a quantum mechanical setting with symmetries. He described three types of independent irreducible ensembles: complex Hermitian, real symmetric, and quaternion self-dual. This article first reviews Dyson’s threefold way from a modern perspective before considering a minimal extension of his setting to incorporate the physics of chiral Dirac fermions and disordered superconductors. In this minimally extended setting, Hilbert space is replaced by Fock space equipped with the anti-unitary operation of particle-hole conjugation, and symmetry classes are in one-to-one correspondence with the large families of Riemannian symmetric spaces.
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Lei, Yuan. Lung Ventilation: Natural and Mechanical. Oxford University Press, 2017. http://dx.doi.org/10.1093/med/9780198784975.003.0003.
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‘Lung Ventilation: Natural and Mechanical’ describes the processes of respiration and lung ventilation, focusing on those issues related directly to mechanical ventilation. The chapter starts by discussing the anatomy and physiology of respiration, and the involvement of the lungs and the entire respiratory system. It continues by introducing the three operating principles of mechanical ventilation. It then narrows its focus to intermittent positive pressure ventilation (IPPV), the operating principle of most modern critical care ventilators, explaining the pneumatic process of IPPV. The chapter ends by comparing natural and mechanical/artificial lung ventilation.
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Coopersmith, Jennifer. The whole of physics. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0008.
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How the Principle of Least Action underlies all physics (all physics that can be reduced to mathematical equations) is explained at a qualitative, semi-popular level. It even applies to smartphones. The domains of classical mechanics, continuum mechanics, materials science, light and electromagnetic waves, special and general relativity (Einstein’s Theory of Gravitation), electrodynamics, quantumelectrodynamics (QED), hydrodynamics, physical chemistry, statistical mechanics, and the quantum world, are examined. It is shown that the Principles of Least Time, Least Resistance, and Maximal Ageing, and Lenz’s Law are, in fact, examples of the Principle of Least Action. It is also shown how Planck’s constant is a measure of “absolute smallness,” and its units are the units of action. Never again, post quantum mechanics, can there be any doubt about the deep significance of action in physics.
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Healey, Richard. Causation in Quantum Mechanics. Edited by Helen Beebee, Christopher Hitchcock, and Peter Menzies. Oxford University Press, 2010. http://dx.doi.org/10.1093/oxfordhb/9780199279739.003.0034.
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There is widespread agreement that quantum mechanics has something radical to teach us about causation. But opinions differ on what this is. Physicists have often taken the central lesson to be that many physical events occur spontaneously, so that a principle of causality is violated whenever an atom emits light, or a uranium nucleus decays, even though nothing that happened beforehand made this inevitable. Feynman urged philosophers to acknowledge that this implication of quantum mechanics undermines the view that causal determinism forms a precondition of scientific inquiry. But while some physicists have denied the implication, most philosophers since Reichenbach have accepted it with alacrity, and sought to develop accounts of causation equally applicable in an indeterministic or a deterministic world.
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Coopersmith, Jennifer. D’Alembert’s Principle. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0005.
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It is explained how the mysterious Principle of Virtual Work in statics is extended to the even more mysterious Principle of d’Alembert’s in dynamics. This is achieved by d’Alembert’s far-sighted stratagem: considering a reversed massy acceleration as an inertial force. A worked example is given (the half-Atwood machine or “black box”). Some counter-intuitive aspects are made intuitive by more examples: the Pluto-Charon system of orbiting planets; Newton’s and then Mach’s explanation of Newton’s bucket. Also, it is demonstrated that the law of the conservation of energy actually follows from d’Alembert’s Principle. The reader is alerted to the astoundingly fundamental nature of d’Alembert’s Principle. It is the cornerstone of classical, relativistic, and quantum mechanics. As Lanczos writes: “All the different principles of mechanics are merely mathematically different formulations of d’Alembert’s Principle”.
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In the Grip of the Distant Universe: The Science of Inertia. World Scientific Publishing Company, 2006.
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22
Eisenberg, Melvin A. Mistranscriptions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199731404.003.0042.
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Chapter 42 concerns mistranscriptions. In a mistranscription case A and B enter into an oral contract and agree that A will transcribe the agreement into writing. Due to A’s negligence the writing mistranscribes the parties’ contract. Both parties then sign the writing, mistakenly believing that it accurately reflects the contract. Mistranscriptions are a special case of mechanical errors: A intends the writing to incorporate the contract, but by virtue of a mechanical error it does not. Mistranscription cases are fairly easy to deal with. The principle that should govern mistranscriptions is the same as the principle that should govern other mechanical errors—a mistranscription should provide a basis for relief to the adversely affected party—specifically, reformation of the writing to make it conform to the oral contract.
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Coopersmith, Jennifer. The Lazy Universe. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.001.0001.
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Action and the Principle of Least Action are explained: what Action is, why the Principle of Least Action works, why it underlies all physics, and what are the insights gained into energy, space, and time. The physical and mathematical origins of the Lagrange Equations, Hamilton’s Equations, the Lagrangian, the Hamiltonian, and the Hamilton-Jacobi Equation are shown. Also, worked examples in Lagrangian and Hamiltonian Mechanics are given. However the aim is to explain physics rather than to give a technical mastery of the subject. Therefore, much of the mathematics is in the appendices. While there is still some mathematics in the main text, the reader may select whether to work through, skim-read, or skip over it: the “story-line” will just about be maintained whatever route is chosen. The work is a much-reduced and simplified version of the outstanding text, “The Variational Principles of Mechanics” written by Cornelius Lanczos in 1949. That work is barely known today, and the present work may be considered as a tiny stepping-stone toward it. A principle that underlies all of physics will have wider repercussions; it is also to be appreciated in an aesthetic sense. It is hoped that this book will lead the reader to the widest possible understanding of the Principle of Least Action. Ideas such as Variational Mechanics, phase space, Fermat’s Principle, and Noether’s Theorem are explained.
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Caparrini, Sandro, and Craig Fraser. Mechanics in the Eighteenth Century. Edited by Jed Z. Buchwald and Robert Fox. Oxford University Press, 2017. http://dx.doi.org/10.1093/oxfordhb/9780199696253.013.13.
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This article focuses on mechanics in the eighteenth century. The publication in 1687 of Isaac Newton’s Mathematical Principles of Natural Philosophy has long been regarded as the event that ushered in the modern period in mathematical physics. The success and scope of the Principia heralded the arrival of mechanics as the model for the mathematical investigation of nature. This subject would be at the cutting edge of science for the next two centuries. This article first provides an overview of the fundamental principles and theorems of mechanics, including the principles of inertia and relativity, before discussing the dynamics of rigid bodies. It also considers the formulation of mechanics by Jean-Baptiste le Rond d’Alembert and Joseph-Louis Lagrange, the statics and dynamics of elastic bodies, and the mechanics of fluids. Finally, it describes major developments in celestial mechanics.
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Zubairy, M. Suhail. Quantum Mechanics for Beginners. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198854227.001.0001.
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Quantum mechanics is a highly successful yet a mysterious theory. Quantum Mechanics for Beginners provides an introduction of this fascinating subject to someone with only a high school background in physics and mathematics. This book, except the last chapter on the Schrödinger equation, is entirely algebra-based. A major strength of this book is that, in addition to the foundation of quantum mechanics, it provides an introduction to the fields of quantum communication and quantum computing. The topics covered include wave–particle duality, the Heisenberg uncertainty relation, Bohr’s principle of complementarity, quantum superposition and entanglement, Schrödinger’s cat, Einstein–Podolsky–Rosen paradox, Bell theorem, quantum no-cloning theorem and quantum copying, quantum eraser and delayed choice, quantum teleportation, quantum key distribution protocols such as BB-84 and B-92, counterfactual communication, quantum money, quantum Fourier transform, quantum computing protocols including Shor and Grover algorithms, quantum dense coding, and quantum tunneling. All these topics and more are explained fully but using only elementary mathematics. Each chapter is followed by a short list of references and some exercises. This book is meant for an advanced high school student and a beginning college student and can be used as a text for a one semester course at the undergraduate level. However it can also be a useful and accessible book for those who are not familiar but want to learn some of the fascinating recent and ongoing developments in areas related to the foundations of quantum mechanics and its applications to quantum communication and quantum computing.
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Mann, Peter. Liouville’s Theorem & Classical Statistical Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0020.
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This chapter returns to the discussion of constrained Hamiltonian dynamics, now in the canonical setting, including topics such as regular Lagrangians, constraint surfaces, Hessian conditions and the constrained action principle. The standard approach to Hamiltonian mechanics is to treat all the variables as being independent; in the constrained case, a constraint function links the variables so they are no longer independent. In this chapter, the Dirac–Bergmann theory for singular Lagrangians is developed, using an action-based approach. The chapter then investigates consistency conditions and Dirac’s different types of constraints (i.e. first-class constraints, second-class constraints, primary constraints and secondary constraints) before deriving the Dirac bracket from simple arguments. The Jackiw–Fadeev constraint formulation is then discussed before the chapter closes with the Güler formulation for a constrained Hamilton–Jacobi theory.
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Cohadon, Pierre-François, Jack Harris, Florian Marquardt, and Leticia Cugliandolo, eds. Quantum Optomechanics and Nanomechanics. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198828143.001.0001.
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The Les Houches Summer School 2015 covered the emerging fields of cavity optomechanics and quantum nanomechanics. Optomechanics is flourishing and its concepts and techniques are now applied to a wide range of topics. Modern quantum optomechanics was born in the late 70s in the framework of gravitational wave interferometry, initially focusing on the quantum limits of displacement measurements. Carlton Caves, Vladimir Braginsky, and others realized that the sensitivity of the anticipated large-scale gravitational-wave interferometers (GWI) was fundamentally limited by the quantum fluctuations of the measurement laser beam. After tremendous experimental progress, the sensitivity of the upcoming next generation of GWI will effectively be limited by quantum noise. In this way, quantum-optomechanical effects will directly affect the operation of what is arguably the world’s most impressive precision experiment. However, optomechanics has also gained a life of its own with a focus on the quantum aspects of moving mirrors. Laser light can be used to cool mechanical resonators well below the temperature of their environment. After proof-of-principle demonstrations of this cooling in 2006, a number of systems were used as the field gradually merged with its condensed matter cousin (nanomechanical systems) to try to reach the mechanical quantum ground state, eventually demonstrated in 2010 by pure cryogenic techniques and a year later by a combination of cryogenic and radiation-pressure cooling. The book covers all aspects—historical, theoretical, experimental—of the field, with its applications to quantum measurement, foundations of quantum mechanics and quantum information. Essential reading for any researcher in the field.
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Nolte, David D. On the Quantum Footpath. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805847.003.0008.
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This chapter shows how the concept of the trajectory of a quantum particle almost vanished in the battle between Werner Heisenberg’s matrix mechanics and Erwin Schrödinger’s wave mechanics. It took Niels Bohr and his complementarity principle of wave-particle duality to cede back some reality to quantum trajectories. However, Schrödinger and Einstein were not convinced and conceived of quantum entanglement to refute the growing acceptance of the Copenhagen Interpretation of quantum physics. Schrödinger’s cat was meant to be an absurdity, but ironically it has become a central paradigm of practical quantum computers. Quantum trajectories took on new meaning when Richard Feynman constructed quantum theory based on the principle of least action, inventing his famous Feynman Diagrams to help explain quantum electrodynamics.
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Heinz, Post, French Steven, and Kamminga Harmke, eds. Correspondence, invariance, and heuristics: Essays in honour of Heinz Post. Dordrecht: Kluwer Academic, 1993.
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(Editor), S. French, and H. Kamminga (Editor), eds. Correspondence, Invariance and Heuristics: Essays in Honour of Heinz Post (Boston Studies in the Philosophy of Science). Springer, 1993.
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31
Phillip, Bricker, and Hughes R. I. G, eds. Philosophical perspectives on Newtonian science. Cambridge, Mass: MIT Press, 1990.
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32
Mercati, Flavio. Barbour–Bertotti Best Matching. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198789475.003.0004.
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Barbour and Bertotti’s Mach–Poincaré Principle can be realized in classical mechanics with a mathematical procedure which was beyond the grasp of Leibniz or Newton, and turns out to be equivalent to modern gauge theory. This is the formulation of a variational principle based on ‘best matching’: one transforms subsequent configurations of the system with the Euclidean group, and by minimizing a certain functional a notion of ‘equilocality’ is established: now it makes sense to say that a particle comes back to the same point at different times.
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Lei, Yuan. Ventilator System Concept. Oxford University Press, 2017. http://dx.doi.org/10.1093/med/9780198784975.003.0004.
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‘Ventilator System Concept’ develops the idea that the equipment for mechanical ventilation is a ventilator system with six essential parts. Using a simple balloon model, it explains the operating principle and composition of a ventilator system, and it describes how the system generates intermittent positive airway pressure. The chapter ends by describing the conditions required for a ventilator system to function properly.
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Mann, Peter. Virtual Work & d’Alembert’s Principle. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0013.
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This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.
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Thurner, Stefan, Rudolf Hanel, and Peter Klimekl. Statistical Mechanics and Information Theory for Complex Systems. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198821939.003.0006.
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Most complex systems are statistical systems. Statsitical mechanics and information theory usually do not apply to complex systems because the latter break the assumptions of ergodicity, independence, and multinomial statistics. We show that it is possible to generalize the frameworks of statistical mechanics and information theory in a meaningful way, such that they become useful for understanding the statistics of complex systems.We clarify that the notion of entropy for complex systems is strongly dependent on the context where it is used, and differs if it is used as an extensive quantity, a measure of information, or as a tool for statistical inference. We show this explicitly for simple path-dependent complex processes such as Polya urn processes, and sample space reducing processes.We also show it is possible to generalize the maximum entropy principle to path-dependent processes and how this can be used to compute timedependent distribution functions of history dependent processes.
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Mann, Peter. Newton’s Three Laws. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0001.
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This chapter introduces Newton’s laws, the Newtonian formulation of mechanics and key concepts such as configuration space and phase space for later development. In 1687, the natural philosopher Sir Isaac Newton published the Principia Mathematica and, with it, sparked the revolutionary ideas key to all branches of classical physics. In this chapter, the system is the object of interest and is considered to be either a single or a collection of generic particles that are not governed by quantum mechanics, for quantum systems do not follow these laws explicitly. Results for systems of particles and conservation laws are presented as the invariance of a given quantity under time evolution. The N-body problem, first integrals, initial value problems and Galilean transformations are all introduced and the Picard iteration and the Verlet algorithm are discussed.
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Rau, Jochen. Introduction. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780199595068.003.0001.
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Statistical mechanics concerns the transition from the microscopic to the macroscopic realm. On a macroscopic scale new phenomena arise that have no counterpart in the microscopic world. For example, macroscopic systems have a temperature; they might undergo phase transitions; and their dynamics may involve dissipation. How can such phenomena be explained? This chapter discusses the characteristic differences between the microscopic and macroscopic realms and lays out the basic challenge of statistical mechanics. It suggests how, in principle, this challenge can be tackled with the help of conservation laws and statistics. The chapter reviews some basic notions of classical probability theory. In particular, it discusses the law of large numbers and illustrates how, despite the indeterminacy of individual events, statistics can make highly accurate predictions about totals and averages.
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T. Michaltsos, George, and Ioannis G. Raftoyiannis, eds. Bridges’ Dynamics. BENTHAM SCIENCE PUBLISHERS, 2012. http://dx.doi.org/10.2174/97816080522021120101.
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Bridges’ Dynamics covers the historical review of research and introductory mathematical concepts related to the structural dynamics of bridges. The e-book explains the theory behind engineering aspects such as 1) dynamic loadings, 2) mathematical concepts (calculus elements of variations, the d’ Alembert principle, Lagrange’s equation, the Hamilton principle, the equations of Heilig, and the δ and H functions), 3) moving loads, 4) bridge support mechanics (one, two and three span beams), 5) Static systems under dynamic loading 6) aero-elasticity, 7) space problems (2D and 3D) and 8) absorb systems (equations governing the behavior of the bridge-absorber system). The e-book is a useful introductory textbook for civil engineers interested in the theory of bridge structures.
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Coopersmith, Jennifer. Antecedents. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0002.
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Early ideas about optimization principles were brought in by an eclectic group of extraordinary thinkers: the Ancients (Hero, and Princess Dido), Fermat with his Principle of Least Time, the Bernoullis, Leibniz, Maupertuis, Euler, and d’Alembert. Also, Stevin was the first to invoke the impossibility of perpetual motion in a proof, and Huygens was the first to put Galilean Relativity to a quantitative test. The Swiss family of mathematical geniuses, the Bernoullis, tackled isoperimetric problems, such as the brachystochrone, and Johann Bernoulli discovered the Principle of Virtual Velocities. The flavour of the eighteenth century is shown in the evocative tale of the König affair, and the correspondence between Daniel Bernoulli and Euler. It is shown how symmetry arguments, leading ultimately to an energy-analysis, were competing with Newton’s force-analysis. The Principle of Least Action and Variational Mechanics, proper, were developed by Lagrange, Hamilton, and Jacobi.
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Coopersmith, Jennifer. The Principle of Virtual Work. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0004.
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The meaning behind the mysterious Principle of Virtual Work is explained. Some worked examples in statics (equilibrium) are given, and the method of Virtual Work is compared and contrasted with the method of Newtonian Mechanics. The meaning of virtual displacements is explained very carefully. They must be ‘small’, happen simultaneously, and do not cause a force, result froma force, or take any time to occur. Counter to intuition, not all the actual displacements can be allowed as virtual displacements. Some examples worked through are: Feynman’s pivoting (cantilever) bar, a “black box,” a weighted spring, a ladder, a capacitor, a soap bubble, and Atwood’s machine. The links between mechanics and geometry are demonstrated, and it is shown how the reaction or constraint forces are always perpendicular to the virtual displacements. Lanczos’s Postulate A and its astounding universality are explained.
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Quantum-Classical Correspondence: Dynamical Quantization and the Classical Limit (The Frontiers Collection). Springer, 2004.
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Mann, Peter. The Stationary Action Principle. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0007.
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This crucial chapter focuses on the stationary action principle. It introduces Lagrangian mechanics, using first-order variational calculus to derive the Euler–Lagrange equation, and the inverse problem is described. The chapter then considers the Ostrogradsky equation and discusses the properties of the extrema using the second-order variation to the action. It then discusses the difference between action functions (of Dirichlet boundary conditions) and action functionals of the extremal path. The different types of boundary conditions (Dirichlet vs Neumann) are elucidated. Topics discussed include Hessian conditions, Douglas’s theorem, the Jacobi last multiplier, Helmholtz conditions, Noether-type variation and Frenet–Serret frames, as well as concepts such as on shell and off shell. Actions of non-continuous extremals are examined using Weierstrass–Erdmann corner conditions, and the action principle is written in the most general form as the Hamilton–Suslov principle. Important applications of the Euler–Lagrange formulation are highlighted, including protein folding.
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Darrigol, Olivier. Constructing Thermal Equilibrium (1866–1871). Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198816171.003.0003.
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This chapter is the first subset of a set of critical summaries Boltzmann’s writings on kinetic-molecular theory. It covers a first period in which he tried to construct the laws of thermal equilibrium, including the existence of the entropy function and the Maxwell–Boltzmann law, by various means including the principle of least action, Maxwell’s collision formula, the ergodic hypothesis, and a procedure of adiabatic variation. This is an immensely fertile period in which Boltzmann introduced several of the basic concepts, problems, and difficulties of modern statistical mechanics.
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Hazony, Yoram, and Eric Schliesser. Newton and Hume. Edited by Paul Russell. Oxford University Press, 2014. http://dx.doi.org/10.1093/oxfordhb/9780199742844.013.28.
Full textAbstract:
Central aspects of Hume’s proposed “system of the sciences” as described in the Treatise are modeled on Newton’s Principia. But, as recent scholarship has suggested, Hume’s Treatise also bears a deeply subversive message with respect to Newtonian science. This chapter offers a revised overview of what Hume takes from Newton and what he rejects: The first part of the chapter argues that in the Treatise Hume adopts a version of Newton’s “analytic and synthetic method” for philosophy, thereby placing a distinctively Newtonian form of explanatory reduction at the center of his own philosophical method. The second part of the chapter, on the other hand, shows that many of the most important aspects of Hume’s argument in Book 1 of the Treatise can be understood as critical of core conceptual and ontological commitments of Newton’s mechanics as developed in the Principia.
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Mashhoon, Bahram. Acceleration Kernel. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.003.0003.
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The phenomenon of spin-rotation coupling provides the key to the determination of the kernel. Imagine an observer rotating in the positive sense about the direction of propagation of an incident plane monochromatic electromagnetic wave of positive helicity. Using the locality postulate, the field as measured by the rotating observer can be determined. If the observer rotates with the same frequency as the wave, the measured radiation field loses its temporal dependence. By a mere rotation, observers could in principle stay at rest with respect to an incident positive-helicity wave. To avoid this possibility, we assume that a basic radiation field cannot stand completely still with respect to an accelerated observer. This basic principle eventually leads to the determination of the kernel and a nonlocal theory of accelerated systems that is in better agreement with quantum mechanics than the standard theory based on the hypothesis of locality.
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Ben-Menahem, Yemima. Causation in Science. Princeton University Press, 2018. http://dx.doi.org/10.23943/princeton/9780691174938.001.0001.
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This book explores the role of causal constraints in science, shifting our attention from causal relations between individual events—the focus of most philosophical treatments of causation—to a broad family of concepts and principles generating constraints on possible change. The book looks at determinism, locality, stability, symmetry principles, conservation laws, and the principle of least action—causal constraints that serve to distinguish events and processes that our best scientific theories mandate or allow from those they rule out. The book's approach reveals that causation is just as relevant to explaining why certain events fail to occur as it is to explaining events that do occur. It investigates the conceptual differences between, and interrelations of, members of the causal family, thereby clarifying problems at the heart of the philosophy of science. The book argues that the distinction between determinism and stability is pertinent to the philosophy of history and the foundations of statistical mechanics, and that the interplay of determinism and locality is crucial for understanding quantum mechanics. Providing a historical perspective, the book traces the causal constraints of contemporary science to traditional intuitions about causation, and demonstrates how the teleological appearance of some constraints is explained away in current scientific theories such as quantum mechanics. The book represents a bold challenge to both causal eliminativism and causal reductionism—the notions that causation has no place in science and that higher-level causal claims are reducible to the causal claims of fundamental physics.
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Byers, Mark. Difficulties of Discovery. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198813255.003.0006.
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The uncertainty of the glyph, reflecting a new commitment to the unpredictability of history and the fallibility of scientific reason, is shown in this chapter to have generated a major avant-garde interest in modern physics, particularly quantum mechanics. The chapter charts cognate developments in Olson’s work and that of Wolfgang Paalen, an Austrian-Mexican painter who had a decisive influence on abstract expressionism through his journal Dyn. Both Olson and Paalen are shown to have turned to post-classical physics—particularly Heisenberg’s ‘uncertainty principle’—as a platform for a new late modernist art that would break with both the political and the aesthetic principles of high modernism.
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Buzon, Frédéric de, and Vincent Carraud. Descartes et les "Principia" II: Corps et mouvement (Philosophies). Presses universitaires de France, 1994.
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