Keplerian Action, Convexity Optimization, and the 4-Body Problem

This chapter uses thought experiments and practical examples to introduce, in a very accessible way, the hard problem of consciousness. Soon, machines may behave like us to pass the Turing test and scientists may succeed in copying and simulating the inner workings of the brain. Will all this take us any closer to solving the mysteries of consciousness? The reader is taken to meet different kind of zombies, the philosophical, the digital, and the inner ones, to understand why many, scientists and philosophers alike, doubt that the mind–body problem will ever be solved.

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Many body dynamics

10.1093/oso/9780198708636.003.0018 ◽

2018 ◽

Author(s):

Joseph F. Boudreau ◽

Eric S. Swanson

Keyword(s):

Statistical Mechanics ◽

Body Problem ◽

Many Body ◽

Complex Objects ◽

Constrained Dynamics ◽

Gravitational Systems ◽

Body Dynamics ◽

System Temperature ◽

Speed Up ◽

Gravitating Systems

Specialized techniques for solving the classical many-body problem are explored in the context of simple gases, more complicated gases, and gravitating systems. The chapter starts with a brief review of some important concepts from statistical mechanics and then introduces the classic Verlet method for obtaining the dynamics of many simple particles. The practical problems of setting the system temperature and measuring observables are discussed. The issues associated with simulating systems of complex objects form the next topic. One approach is to implement constrained dynamics, which can be done elegantly with iterative methods. Gravitational systems are introduced next with stress on techniques that are applicable to systems of different scales and to problems with long range forces. A description of the recursive Barnes-Hut algorithm and particle-mesh methods that speed up force calculations close out the chapter.

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Brute Necessity and the Mind–Body Problem

10.1093/oso/9780198758600.003.0005 ◽

2018 ◽

Author(s):

James Van Cleve

Keyword(s):

Body Problem ◽

Necessary Truths ◽

Mind Body Problem ◽

The Mind

In a growing number of papers one encounters arguments to the effect that certain philosophical views are objectionable because they would imply that there are necessary truths for whose necessity there is no explanation. For short, they imply that there are brute necessities. Therefore, the arguments conclude, the views in question should be rejected in favor of rival views under which the necessities would be explained. This style of argument raises a number of questions. Do necessary truths really require explanation? Are they not paradigms of truths that either need no explanation or automatically have one, being in some sense self-explanatory? If necessary truths do admit of explanation or even require it, what types of explanation are available? Are there any necessary truths that are truly brute? This chapter surveys various answers to these questions, noting their bearing on arguments from brute necessity and arguments concerning the mind–body problem.

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The post-Newtonian approximation

10.1093/oso/9780198786399.003.0052 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

General Relativity ◽

Gravitational Field ◽

Body Problem ◽

Equations Of Motion ◽

Two Body Problem ◽

Newtonian Approximation ◽

Actual Motion ◽

Law Of Attraction ◽

Universal Law ◽

Two Bodies

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

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The two-body problem: an effective-one-body approach

10.1093/oso/9780198786399.003.0056 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Body Problem ◽

Equations Of Motion ◽

Reaction Force ◽

Kerr Black Hole ◽

Radiation Reaction ◽

Spherically Symmetric ◽

Geodesic Motion ◽

Two Body Problem ◽

Symmetric Spacetime ◽

Radiation Reaction Force

This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.

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QLB for Quantum Many-Body and Quantum Field Theory

10.1093/oso/9780199592357.003.0033 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Quantum Mechanics ◽

Quantum Computing ◽

Single Particle ◽

Body Problem ◽

Three Dimensions ◽

Quantum Field ◽

Many Body ◽

Quantum Particles

Chapter 32 expounded the basic theory of quantum LB for the case of relativistic and non-relativistic wavefunctions, namely single-particle quantum mechanics. This chapter goes on to cover extensions of the quantum LB formalism to the overly challenging arena of quantum many-body problems and quantum field theory, along with an appraisal of prospective quantum computing implementations. Solving the single particle Schrodinger, or Dirac, equation in three dimensions is a computationally demanding task. This task, however, pales in front of the ordeal of solving the Schrodinger equation for the quantum many-body problem, namely a collection of many quantum particles, typically nuclei and electrons in a given atom or molecule.

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