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SYSTEMS OF ODEs: MANY-BODY PROBLEMS, NONLINEAR HARMONIC OSCILLATORS

Francesco Calogero

in Isochronous Systems

In Chapter 4—the longer one in this book—a lemma is first introduced and several isochronous systems of ODEs encompassed by it are treated. One-, two-, three- and multi-dimensional isochronous ... More

In Chapter 4—the longer one in this book—a lemma is first introduced and several isochronous systems of ODEs encompassed by it are treated. One-, two-, three- and multi-dimensional isochronous systems of ODEs—many of them interpretable as many-body models—are then discussed, including several integrable and solvable variants of the “goldfish” many-body problem, nonlinear oscillators models, and two Hamiltonian systems.Less

ISOCHRONOUS HAMILTONIAN SYSTEMS ARE NOT RARE

Francesco Calogero

in Isochronous Systems

In Chapter 5 various tricks are introduced, transforming Hamiltonian systems into isochronous Hamiltonian systems. The possibility to apply this procedure to large classes of Hamiltonian systems ... More

In Chapter 5 various tricks are introduced, transforming Hamiltonian systems into isochronous Hamiltonian systems. The possibility to apply this procedure to large classes of Hamiltonian systems justifies the statement that isochronous Hamiltonian systems are not rare. Several examples are discussed. A (very recent) variant of this approach transforms a very large class of (real, autonomous) Hamiltonian systems into Ω-modified isochronous (also real and autonomous)Hamiltonian systems: for instance, it transforms the most general nonrelativistic many-body problem with arbitrary potentials (only restricted to be translation-invariant) into an entirely isochronous many-body problem—completely periodic (in all degrees of freedom) with (arbitrary) period T = 2π/Ω for arbitrary initial data—but behaving, in its center-of-mass frame, essentially as the original system (which might be chaotic) over time intervals short with respect to the period T.Less

ASYMPTOTICALLY ISOCHRONOUS SYSTEMS

Francesco Calogero

in Isochronous Systems

In Chapter 6—reporting very recent findings (although the main idea is not new)—autonomous dynamical systems are treated whose generic solutions approach asymptotically (at large time) isochronous ... More

In Chapter 6—reporting very recent findings (although the main idea is not new)—autonomous dynamical systems are treated whose generic solutions approach asymptotically (at large time) isochronous evolutions, namely, all their dependent variables tend asymptotically to functions periodic with the same, fixed period. Two examples—associated to somewhat different mechanisms—are then discussed: the first is an integrable, indeed solvable many-body model of goldfish type, the second a rather large class of, generally nonintegrable, many-body problems.Less

INTRODUCTION

Francesco Calogero

in Isochronous Systems

In the introductory Chapter 1 a few representative instances of isochronous dynamical systems are tersely reviewed.

MODELS OF MANY-CHAIN SYSTEMS

Glenn H. Fredrickson

in The Equilibrium Theory of Inhomogeneous Polymers

This chapter discusses methods for converting the many body problem in interacting polymer fluids to a statistical field theory. Auxiliary field transforms are introduced, that decouple interactions ... More

This chapter discusses methods for converting the many body problem in interacting polymer fluids to a statistical field theory. Auxiliary field transforms are introduced, that decouple interactions among polymer species and reduce the problem to that of individual polymers interacting with one or more fluctuating fields — the subject of Chapter 3. Examples of models for a variety of systems are enumerated, including polymer solutions, blends, block and graft copolymers, polyelectrolytes, liquid crystalline polymers, and polymer brushes.Less

Applications of path integrals

Efstratios Manousakis

in Practical Quantum Mechanics: Modern Tools and Applications

This chapter discusses some applications of path integrals. First, it illustrates how their qualitative use can yield insights. As an example, it applies it to the so-called Bohm–Aharonov effect. ... More

This chapter discusses some applications of path integrals. First, it illustrates how their qualitative use can yield insights. As an example, it applies it to the so-called Bohm–Aharonov effect. Then, it discusses the generalization of the path-integral formalism for the so-called many-body problem. It only treats distinguishable particles (what we call Boltzmannons) in this chapter. The book will discuss indistinguishable particles in a future chapter. This chapter will also discuss the formulation of quantum statistical mechanics of Boltzmannons.Less

Introduction

Andrew Zangwill

in A Mind Over Matter: Philip Anderson and the Physics of the Very Many

This chapter provides an overview of Anderson’s career and contrasts his speciality, the physics of the very many (solid-state physics), with the areas of physics that tend to appear in popular ... More

This chapter provides an overview of Anderson’s career and contrasts his speciality, the physics of the very many (solid-state physics), with the areas of physics that tend to appear in popular media—the physics of the very small (particle physics) and the physics of the very distant (astrophysics and cosmology). It compares Anderson’s physics skills to those of a chess grandmaster. The number of pieces (atoms and electrons) is so large that merely knowing the microscopic rules of the game is not enough to gain real understanding. There is a focus on the big ideas Anderson brought to the table—symmetry breaking, emergence, and complexity—and also his great interest in the cultural and political aspects of physics. The goal of the book is to help readers understand the magician-like skills Anderson brought to theoretical physics and the effect these had on his students, coworkers, community, and on scientific enterprise.Less