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

Title......Page 1
Copyright......Page 2
Foreword......Page 3
Preface......Page 8
Contents......Page 9
1.1 Newton\'s equation in one, two and three dimensions......Page 15
1.2 Hamiltonian systems - Integrable systems......Page 20
1.N. Notes to Chapter 1......Page 30
2.1 The Lax pair technique......Page 31
2.1.1 A convenient representation. The functional equation (*)......Page 37
2.1.2 A simple solution of the functional equation (*)......Page 40
2.1.3 N particles on the fine, interacting pairwise via repulsive forces......Page 41
2.1.4 General solution of the functional equation (*). Integrable many-body model with elliptic interactions......Page 61
2.1.5 N particles on the line interacting pairwise via a repulsive hyperbolic force. Technique of solution OP......Page 67
2.1.6 N particles on the circle interacting pairwise via a trigonometric force......Page 75
2.1.7 Various tricks: changes of variables, particles of different types, duplications, infinite duplications.........Page 77
2.1.8 Another convenient representation for the Lax pair. The functional equation (**)......Page 94
2.1.9 A simple solution of the functional equation (**)......Page 99
2.1.10 N particles on the line, interacting pairwise via forces equal to twice the product of their velocities.........Page 104
2.1.11 General solution of the functional equation (**)......Page 113
2.1.12 The many-body problem of Ruijsenaars and Schneider (RS)......Page 124
2.1.13 Various tricks: changes of variables, duplications, infinite duplications, reductions to \"nearest-neighbor\"\' forces.........Page 140
2.1.14 Another Lax pair corresponding to a Hamiltonian many-body problem on the line. The functional equation (***)......Page 149
2.1.15 A simple solution of the functional equation (***), and the corresponding Hamiltonian many-body problem on the line......Page 152
2.1.16 A nonanalytic solution of the functional equation (***), and the corresponding Hamiltonian many-body problem......Page 165
2.2 Another exactly solvable Hamiltonian problem......Page 173
2.3 Many-body problems on the line related to the motion of the zeros of solutions of linear partial differential equations........Page 177
2.3.1 A nonlinear transformation: relationships between the coefficients and the zeros of a polynomial......Page 178
2.3.2 Some formulas for a polynomial and its derivatives, in terms of its coefficients and its zeros......Page 179
2.3.3 Many-body problems on the line solvable via the identification of their motions with those of the zeros.........Page 181
2.3.4 Examples......Page 188
2.3.5 Trigonometric extension......Page 211
2.3.6 Further extension......Page 217
2.4 Finite-dimensional representations of differential operators, Lagrangian interpolation, and all that......Page 242
2.4.1 Finite-dimensional matrix representations of differential operators......Page 243
2.4.2 Connection with Lagrangian interpolation......Page 249
2.4.3 Algebraic approach......Page 254
2.4.4 The finite-dimensional (matrix) algebra of raising and lowering operators, and its realizations......Page 267
2.4.5 Remarkable matrices and identities......Page 278
2.5 Many-body problems on the line solvable via techniques of exact Lagrangian interpolation......Page 293
2.N Notes to Chapter 2......Page 318
3.1 Generalized formulation of Lagrangian interpolation, in spaces of arbitrary dimensions......Page 324
3.1.1 Finite-dimensional representation of the operator of differentiation......Page 329
3.1.2 Examples......Page 343
3.2 N -body problems in spaces of one or more dimensions......Page 368
3.2.1 One-dimensional examples......Page 381
3.2.2 Two-dimensional examples (in the plane)......Page 402
3.2.3 Few-body problems in ordinary (3 -dimensional) space......Page 413
3.2.4 M -body problems in M-dimensional space, or M^2-body problems in one-dimensional space......Page 417
3.3 First-order evolution equations and partially solvable N -body problems with velocity-independent forces......Page 421
3.N Notes to Chapter 3......Page 428
4 SOLVABLE AND/OR INTEGRABLE MANY-BODY PROBLEMS IN THE PLANE,OBTAINED BY COMPLEXIFICATION......Page 429
4.1 How to obtain by complexification rotation-invariant many-body models in the plane from certain many-body problems.........Page 430
4.2 Example: a family of solvable many-body problems in the plane......Page 440
4.2.1 Origin of the model and technique of solution......Page 441
4.2.2 The generic model; behavior in the remote past and future......Page 444
4.2.3 Some special cases: models with a limit cycle, models with confined and periodic motions, Hamiltonian models.........Page 447
4.2.4 The simplest model: explicit solution (the game of musical chairs), Hamiltonian structure......Page 460
4.2.5 The simplest model featuring only completely periodic motions......Page 465
4.2.6 First-order evolution equations, and a partially solvable many-body problem with velocity-independent forces.........Page 468
4.3 Examples: other families of solvable many-body problems in the plane......Page 471
4.3.1 A rescaling-invariant solvable one-dimensional many-body problem......Page 474
4.3.2 A rescaling- and translation-invariant solvable one-dimensional many-body problem......Page 479
4.3.3 Another rescaling-invariant solvable one-dimensional many-body problem......Page 481
4.4 Survey of solvable and/or integrable many-body problems in the plane obtained by complexification......Page 483
4.4.1 Example one......Page 484
4.4.2 Example two......Page 486
4.4.3 Example three......Page 488
4.4.4 Example four......Page 491
4.4.5 Example five......Page 493
4.4.6 Example six......Page 494
4.4.7 Example seven......Page 497
4.4.8 Example eight......Page 501
4.4.9 Example nine......Page 503
4.4.10 A Hamiltonian example......Page 505
4.5 A many-rotator, possibly nonintegrable, problem in the plane, and its periodic motions......Page 506
4.N Notes to Chapter 4......Page 521
5 MANY-BODY SYSTEMS IN ORDINARY (THREE-DIMENSIONAL) SPACE: SOLVABLE, INTEGRABLE, LINEARIZABLE PROBLEMS......Page 523
5.1 A simple example: a solvable matrix problem, and the corresponding one-body problem in three-dimensional space......Page 524
5.2 Another simple example: a linearizable matrix problem, and the corresponding one-body problem in three-dimensional space......Page 532
5.2.1 Motion of a magnetic monopole in a central electric field......Page 547
5.2.2 Motion of a magnetic monopole in a central Coulomb field......Page 555
5.2.3 Solvable cases of the (2 x 2)-matrix evolution equation......Page 562
5.3 Association, complexification, multiplication: solvable few and many-body problems obtained from the previous ones......Page 565
5.4 A survey of matrix evolution equations amenable to exact treatments......Page 581
5.4.1 A class of linearizable matrix evolution equations......Page 582
5.4.2 Some integrable matrix evolution equations related to the non Abelian Toda lattice......Page 597
5.4.3 Some other matrix evolution equations amenable to exact treatments......Page 602
5.4.4 On the integrability of the matrix evolution equation U = f(U)......Page 610
5.5 Parametrization of matrices via three-vectors......Page 617
5.6 A survey of N -body systems in three-dimensional space amenable to exact treatments......Page 625
5.6.1 Few-body problems of Newtonian type......Page 626
5.6.2 Few-body problems of Hamiltonian type......Page 638
5.6.3 Many-body problems of Newtonian type......Page 640
5.6.4 Many-body problems of Hamiltonian type......Page 653
5.6.5 Many-body problems in multidimensional space with velocity-independent forces: integrable unharmonic (\"quartic\").........Page 657
5.7 Outlook......Page 673
5.N Notes to Chapter 5......Page 674
Appendix A: Elliptic functions......Page 675
Appendix B: Functional equations......Page 687
Appendix C: Hermite polynomials: zeros, determinantal representations......Page 697
Appendix D: Remarkable matrices and related identities......Page 701
A irspendix E: Lagrangian approximation for eigenvalue problems in one and more dimensions......Page 715
Appendix F: Some theorems of elementary geometry in multidimensions......Page 720
Appendix G: Asymptotic behavior of the zeros of a polynomial whose coefficients diverge exponentially......Page 735
Appendix H: Some formulas for Pauli matrices and three-vectors......Page 744
References......Page 746