Classical many-body problems amenable to exact treatments : solvable and/or integrable and/or linearizable ... in one-, two-, and three- dimensional space

✍ Scribed by Calogero, Francesco

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 747
• Series: Lecture notes in physics. New series m, Monographs ; m66
• Category: Library

✦ Synopsis

"This book focuses on exactly treatable classical (i.e. non-quantal non-relativistic) many-body problems, as described by Newton's equation of motion for mutually interacting point particles. Most of the material is based on the author's research and is published here for the first time in book form. One of the main novelties is the treatment of problems in two- and three-dimensional space. Many related techniques are presented, e.g. the theory of generalized Lagrangian-type interpolation in higher-dimensional spaces. This book is written for students as well as for researchers; it works out detailed examples before going on to treat more general cases. Many results are presented via exercises, with clear hints pointing to their solutions."--Jacket. Read more...

Abstract:

This text focuses on exactly treatable classical non-quantal, non-relativistic many-body problems mainly focusing on point particles determined using Newton's equations of motion. Many techniques are presented, such as techniques using Lagrangian-type approximations in higher-dimensional spaces. Read more...

✦ Table of Contents

Content: 1. Classical (nonquantal, nonrelativistic) many-body problems --
1.1. Newton's equation in one, two and three dimensions --
1.2. Hamiltonian systems --
Integrable systems --
2. One-dimensional systems. Motions on the line and on the circle --
2.1. The Lax pair technique --
2.2. Another exactly solvable Hamiltonian problem --
2.3. Many-body problems on the line related to the motion of the zeros of solutions of linear partial differential equations in 1+1 variables (space+time) --
2.4. Finite-dimensional representations of differential operators, Lagrangian interpolation, and all that --
2.5. Many-body problems on the line solvable via techniques of exact Lagrangian interpolation --
3. N-body problems treatable via techniques of exact Lagrangian interpolation in spaces of one or more dimensions --
3.1. Generalized formulation of Lagrangian interpolation, in spaces of arbitrary dimensions --
3.2. N-body problems in spaces of one or more dimensions --
3.3. First-order evolution equations and partially solvable N-body problems with velocity-independent forces --
4. Solvable and/or integrable many-body problems in the plane, obtained by complexification --
4.1. How to obtain by complexification rotation-invariant many-body models in the plane from certain many-body problems on the line --
4.2. Example: a family of solvable many-body problems in the plane --
4.3. Examples: other families of solvable many-body problems in the plane --
4.4. Survey of solvable and/or integrable many-body problems in the plane obtained by complexification --
4.5. A many-rotator, possibly nonintegrable, problem in the plane, and its periodic motions --
4.6. Outlook --
5. Many-body systems in ordinary (three-dimensional) space: solvable, integrable, linearizable problems --
5.1. A simple example: a solvable matrix problem, and the corresponding one-body problem in three-dimensional space --
5.2. Another simple example: a linearizable matrix problem, and the corresponding one-body problem in three-dimensional space --
5.3. Association, complexification, multiplication: solvable few and many-body problems obtained from the previous ones --
5.4. A Survey of matrix evolution equations amenable to exact treatments --
5.5. Parametrization of matrices via three-vectors --
5.6. A survey of N -body systems in three-dimensional space amenable to exact treatments --
5.7. Outlook --
App. A. Elliptic functions --
App. B. Functional equations --
App. C. Hermite polynomials: zeros, determinantal representations --
App. D. Remarkable matrices and related identities --
App. E. Lagrangian approximation for eigenvalue problems in one and more dimensions --
App. F. Some theorems of elementary geometry in multidimensions --
App. G. Asymptotic behavior of the zeros of a polynomial whose coefficients diverge exponentially --
App. H. Some formulas for Pauli matrices and three-vectors.

✦ Subjects

Физика;Механика;