The Problem of Integrable Discretization: Hamiltonian Approach

The Problem of Integrable Discretization: Hamiltonian Approach

by Yuri B. Suris

Paperback(Softcover reprint of the original 1st ed. 2003)

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Product Details

ISBN-13: 9783034894043
Publisher: Birkhäuser Basel
Publication date: 11/05/2012
Series: Progress in Mathematics , #219
Edition description: Softcover reprint of the original 1st ed. 2003
Pages: 21
Product dimensions: 6.10(w) x 9.25(h) x (d)

Table of Contents

I General Theory.- 1 Hamiltonian Mechanics.- 1.1 The problem of integrable discretization.- 1.2 Poisson brackets and Hamiltonian flows.- 1.3 Symplectic manifolds.- 1.4 Poisson submanifolds and symplectic leaves.- 1.5 Dirac bracket.- 1.6 Poisson reduction.- 1.7 Complete integrability.- 1.8 Bi-Hamiltonian systems.- 1.9 Lagrangian mechanics on ?N.- 1.10 Lagrangian mechanics on TP and on P × P.- 1.11 Lagrangian mechanics on Lie groups.- 1.11.1 Continuous time case.- 1.11.2 Discrete time case.- 1.12 Invariant Lagrangians and Lie-Poisson bracket.- 1.12.1 Continuous time case.- 1.12.2 Discrete time case.- 1.13 Lagrangian reduction and Euler-Poincaré equations.- 1.13.1 Continuous time case.- 1.13.2 Discrete time case.- A Appendix: Gradients, vector fields, and other notation.- B Appendix: Lie groups and Lie algebras.- 1.14 Bibliographical remarks.- 2 R-matrix Hierarchies.- 2.1 Introduction.- 2.2 Lie-Poisson brackets.- 2.2.1 General construction.- 2.2.2 Tensor notation.- 2.2.3 Examples.- 2.3 Linear r-matrix structure.- 2.3.1 General construction.- 2.3.2 Tensor notation.- 2.3.3 Examples of R-operators and r-matrices.- 2.4 Generalized linear r-matrix structure.- 2.5 Quadratic r-matrix structure.- 2.5.1 General construction.- 2.5.2 Tensor notation.- 2.5.3 Example.- 2.6 Poisson brackets on direct products.- 2.6.1 General construction.- 2.6.2 Tensor notation.- 2.6.3 Poisson properties of the monodromy map.- 2.7 R-operators from splitting g = g+? g-.- 2.8 Bäcklund transformations.- 2.9 Recipe for integrable discretization.- A Appendix: Bäcklund-Darboux transformation for KdV.- 2.10 Bibliographical remarks.- II Lattice Systems.- 3 Toda Lattice.- 3.1 Introduction.- 3.2 Tri-Hamiltonian structure.- 3.3 Basic algebras and operators.- 3.3.1 Open-end case.- 3.3.2 Periodic case.- 3.4 Lax representation.- 3.5 Linear r-matrix structure.- 3.6 Quadratic r-matrix structure.- 3.7 2 × 2 Lax representation.- 3.8 Discretization of the Toda lattice.- 3.9 Localizing changes of variables.- 3.10 Local equations of motion for dTL.- 3.11 Second Toda flow and its discretization.- 3.12 Local equations of motion for dTL2.- 3.13 Third Toda flow and its discretization.- 3.14 Local equations of motion for dTL3.- 3.15 Modified Toda lattice.- 3.16 Discretization of MTL.- 3.17 Local equations of motion for dMTL.- 3.18 Second modification of TL.- 3.19 Discretization of M2TL.- 3.20 Local equations of motion for dM2TL.- 3.21 Third modification of TL.- A Appendix: Miura transformations for KdV.- 3.22 Bibliographical remarks.- 4 Volterra Lattice.- 4.1 Introduction.- 4.2 Bi-Hamiltonian structure.- 4.3 Lax representation.- 4.4 r-matrix structure.- 4.5 Discretization.- 4.6 Local equations of motion for dVL.- 4.7 Second flow of the Volterra hierarchy.- 4.8 Local discretization of VL2.- 4.9 Local discretization of KdV.- 4.10 Modified Volterra lattice.- 4.11 Discretization of MVL.- 4.12 Local equations of motion for dMVL.- 4.13 Different forms of MVL and dMVL.- 4.14 Particular case € ? ? of MVL.- 4.15 Second modification of VL.- 4.16 Factorizations and the two-field form of VL.- 4.17 Lax representation in g ? g.- 4.18 Quadratic r-matrix structure in g ? g.- 4.19 Discretization of the two-field VL.- 4.20 Local equations for the two-field dVL.- 4.21 Two-field versions of VL2 and dVL2.- 4.22 Two-field modified Volterra lattice.- 4.23 Discretization of the two-field MVL.- 4.24 Local equations for the two-field dMVL.- A Appendix: Tower of modifications of VL à la Yamilov.- 4.26 Bibliographical remarks.- 5 Newtonian Equations of the Toda Type.- 5.1 Introduction.- 5.2 Exponential form of the Toda lattice.- 5.3 Dual Toda lattice.- 5.4 Modified exponential Toda lattice.- 5.5 Parametrizing the linear-quadratic bracket.- 5.6 Parametrizing the cubic-quadratic bracket.- 5.7 Parametrizing the cubic bracket I.- 5.8 Parametrizing the cubic bracket II.- 5.9 Parametrizing the cubic bracket III.- 5.10 Newtonian equations for TL2.- 5.11 Bibliographical remarks.- 6 Relativistic Toda Lattice.- 6.1 Introduction.- 6.2 The first Lax representation of RTL(?).- 6.3 Linear r-matrix for the first Lax representation.- 6.4 Quadratic r-matrix for the first Lax representation.- 6.5 Tri-Hamiltonian structure of RTL(?).- 6.6 The second Lax representation of RVL(?).- 6.7 Linear r-matrix for the second Lax representation.- 6.8 Quadratic r-matrix for the second Lax representation.- 6.9 2 × 2 Lax representations.- 6.10 Discretization of the flow RTL+(?).- 6.11 Localizing change of variables for dRTL+(?).- 6.12 Discretization of the flow RTL-(?).- 6.13 Localizing change of variables for dRTL-(?).- 6.14 Modified relativistic Toda lattice MRTL(?; €).- 6.15 Different forms of MRTL(?; €).- 6.15.1 Change of variables corresponding to M) (?; €).- 6.15.2 Change of variables corresponding to MZ+) (?; €).- 6.16 Lax representations of MRTL(?; €).- 6.16.1 Lax representation corresponding to M) (?; €).- 6.16.2 Lax representation corresponding to MZ+) (?; €).- 6.17 r-matrix interpretation of MRTL(?; €).- 6.18 Discretization of MRTL+(?; €).- 6.18.1 Discretization based on the first Lax representation.- 6.18.2 Discretization based on the second Lax representation.- 6.19 Localizing change of variables for dMRTL+(?; €).- 6.20 Discretization of MRTL_ (?; €).- 6.21 Bibliographical remarks.- 7 Relativistic Volterra Lattice.- 7.1 Introduction.- 7.2 Quadratic invariant Poisson bracket of RVL(?).- 7.3 Cubic invariant Poisson bracket of RVL(?).- 7.4 Auto-transformation of RVL(?).- 7.5 The first Lax representation of RVL(?).- 7.6 Quadratic r-matrix for the first Lax representation.- 7.7 The second Lax representation of RVL(?).- 7.8 The third Lax representation of RVL(?).- 7.9 Quadratic r-matrix for third Lax representation.- 7.10 Discretization of RVL+(?).- 7.11 Localizing change of variables for dRVL+(?).- 7.12 Discretization of RVL-(?).- 7.13 Localizing change of variables for dRVL-(?).- 7.14 Modified relativistic Volterra lattice.- 7.15 Discretization of MRVL+(?; €).- 7.16 Appendix: selected results for M1-version of RVL(?).- 7.16.1 The flow RVL+(?) and its discretization.- 7.16.2 The flow RVL-(?) and its discretization.- 7.16.3 The flow MRVL+(?; €) and its discretization.- 7.17 Bibliographical remarks.- 8 Newtonian Equations of the Relativistic Toda Type.- 8.1 Introduction.- 8.2 Parametrizing the linear-quadratic bracket.- 8.2.1 Systems RTL+(?), dRTL+(?).- 8.2.2 Systems RTL-(?), dRTL-(?).- 8.3 Parametrizing the linear bracket.- 8.3.1 Systems RTL+(?), dRTL+(?).- 8.3.2 Systems RTL-(?), dRTL-(?).- 8.4 Dual linear parametrization.- 8.4.1 Systems RTL+(?), dRTL+(?).- 8.4.2 Systems RTL-(?), dRTL-(?).- 8.5 Parametrizing the quadratic bracket.- 8.5.1 Systems RVL+(?), dRVL+(?).- 8.5.2 Systems RVL-(?), dRVL-(?).- 8.6 Parametrizing the linear-quadratic bracket II.- 8.6.1 Systems MRTL+(?; €), dMRTL+(?; €).- 8.6.2 Systems MRTL-(?; €), dMRTL-(?; €).- 8.7 Parametrizing the cubic-quadratic bracket.- 8.8 Parametrizing the cubic bracket I.- 8.9 Parametrizing the cubic bracket II.- 8.10 Parametrizing the cubic bracket III.- 8.11 Bibliographical remarks.- 9 Explicit Discretizations for Toda Systems.- 9.1 Introduction.- 9.2 Explicit discretization for TL.- 9.3 Explicit discretization for MTL(€).- 9.4 Explicit discretization for VL.- 9.5 Explicit discretization for MVL(€).- 9.6 Explicit dRTL+(h) from implicit dTL.- 9.7 Explicit dRVL+(h) from implicit dVL.- 9.8 Bibliographical remarks.- 10 Explicit Discretizations of Newtonian Toda Systems.- 10.1 Introduction.- 10.2 Parametrizing special linear-quadratic bracket.- 10.3 Parametrizing the linear bracket.- 10.4 Dual parametrization of the linear bracket.- 10.5 Parametrizing the quadratic bracket.- 10.6 Parametrizing general linear-quadratic bracket.- 10.7 Parametrizing the cubic-quadratic bracket.- 10.8 Parametrizing the cubic bracket. I.- 10.9 Parametrizing the cubic bracket. II.- 10.10 Parametrizing the cubic bracket. III.- 10.11 Bibliographical remarks.- 11 Bruschi-Ragnisco Lattice.- 11.1 Introduction.- 11.2 Bi-Hamiltonian structure.- 11.3 General construction.- 11.4 Orbit interpretation.- 11.5 Discretization.- 11.6 Newtonian equations of motion.- 11.7 Bibliographical remarks.- 12 Multi-field Toda-like Systems.- 12.1 Introduction.- 12.2 Multi-field analog of the Toda lattice.- 12.3 Linear r-matrix structure for TLm+1.- 12.4 Quadratic r-matrix structure for TLm+1.- 12.5 Discretization of TLm+1.- 12.6 Localizing change of variables for dTLm+1.- 12.7 Example: TL3.- 12.8 Multi-field analog of the modified Toda lattice.- 12.9 Quadratic r-matrix structure for MTLm+1(€).- 12.10 Discretization of MTLm+i (€).- 12.11 Localizing change of variables for dMTLm+i (E).- 12.12 Bibliographical remarks.- 13 Multi-field Relativistic Toda Systems.- 13.1 Introduction.- 13.2 Multi-field analog of RTL: first version.- 13.3 Linear r-matrix structure for RTLm+1(?).- 13.4 Quadratic bracket for RTLm+1(?).- 13.5 Introducing the gauge transformed hierarchy.- 13.6 Multi-field RTL: second version.- 13.7 Quadratic r-matrix structure for RTLm+1(?).- 13.8 Example: RTL3(?).- 13.9 Discretization of RTLm++1(?).- 13.10 Localizing change of variables for dRTL~++l (?).- 13.11 Local discretization for RTL, +1(?).- 13.12 Bibliographical remarks.- 14 Belov-Chaltikian Lattices.- 14.1 Introduction.- 14.2 Bi-Hamiltonian structure and Lax representation.- 14.3 Discretization of BCLm.- 14.4 Modified BCLm.- 14.5 Discretization of MBCLm(€).- 14.6 Localizing change of variables for dMBCLm( €).- 14.7 Relativistic deformation of BCLm.- 14.8 A gauge connection between MBCLm( €) and RBCLm(?).- 14.9 Discretization of RBCLm(?).- 14.10 Example: Volterra lattice as BCL1.- 14.11 Example: BCL2.- 14.12 Bibliographical remarks.- 15 Multi-field Volterra-like Systems.- 15.1 Introduction.- 15.2 Multi-field analog of the Volterra lattice.- 15.3 Quadratic r-matrix structure for VLm.- 15.4 Discretization of VLm.- 15.5 Localizing change of variables for dVLm.- 15.6 Example 1: VL3, three-field analog of Volterra lattice.- 15.7 A further generalization of VLm.- 15.8 Quadratic r-matrix structure for VLm(?).- 15.9 Discretization of VLm(?).- 15.10 Localizing change of variables for dVLm(?).- 15.11 The case of the signature ? = (+1, -1,…, -1).- 15.11.1 Lax representation and Hamiltonian structure.- 15.11.2 Discretization.- 15.11.3 Localizing change of variables.- 15.11.4 Miura relation to the Belov-Chaltikian lattices.- 15.12 Example 2: ? = (+1, -1, -1).- 15.13 Example 3: ? = (+1, +1, -1).- 15.14 Bibliographical remarks.- 16 Multi-field Relativistic Volterra Systems.- 16.1 Introduction.- 16.2 The RVLm(?; ?) hierarchy: first construction.- 16.3 Introducing the gauge transformed hierarchy.- 16.4 The RVLm( ?; ?) hierarchy: second construction.- 16.5 Discretization of RVI4,+) (?; ?).- 16.6 Localizing change of variables for dRVL,~,,+) (?; ?).- 16.7 Particular case h = a: explicit discretizations.- 16.8 The case of the signature v = (+1, +1,¡­, +1).- 16.8.1 Equations of motion and Hamiltonian structure.- 16.8.2 Discretization.- 16.8.3 Explicit discretization.- 16.8.4 Example: RVL3+“ (?), the three-field analog of the relativistic Volterra lattice.- 16.9 The case of the signature ? = (+1, ¡ª1,¡­, ¡ª1).- 16.9.1 Equations of motion and Hamiltonian structure.- 16.9.2 Discretization.- 16.9.3 Example: RVL3+“ (?; ?) with ? = (+1, ¡ª1, ¡ª1).- 16.10 Explicit dVL.m from the RVLmhierarchy.- 16.11 Bibliographical remarks.- 17 Bogoyavlensky Lattices.- 17.1 Introduction.- 17.2 Lax representations.- 17.3 Quadratic r-matrix structure of BL1(m).- 17.4 Quadratic r-matrix structure of BL2(p) and BL3(p).- 17.5 Examples of Hamiltonian structures.- 17.5.1 Lattice BL2(p), p > 1.- 17.5.2 Lattice BL3(p), p ? 2.- 17.6 Discretization of the lattice BL1(m).- 17.7 Discretization of the lattice BL2(p).- 17.8 Discretization of the lattice BL3 (p).- 17.9 Modified Volterra lattice.- 17.10 Alternative approach to BL1(m).- 17.11 Alternative approach to BL2(p).- 17.12 Alternative approach to BL3 (p).- 17.13 Bibliographical remarks.- 18 Ablowitz-Ladik Hierarchy.- 18.1 Introduction.- 18.2 AKNS hierarchy.- 18.3 Ablowitz-Ladik hierarchy.- 18.4 Non-local difference schemes.- 18.4.1 Difference schemes for NLS.- 18.4.2 Difference schemes for MKdV.- 18.5 Elementary flows of the AL hierarchy.- 18.6 Local discretizations for.F1.- 18.7 Symplectic properties.- 18.8 Local discretizations for NLS.- 18.9 Local discretizations for F 2.- 18.10 Local discretizations for MKdV.- 18.11 Connection with relativistic Toda lattice.- 18.12 Bibliographical remarks.- III Systems of Classical Mechanics.- 19 Peakons System.- 19.1 Introduction.- 19.2 Lax representation and r-matrix.- 19.3 Discretization.- 19.4 Lagrangian interpretation.- 19.5 Bibliographical remarks.- 20 Standard-like Discretizations.- 20.1 Introduction.- 20.2 Integrable scalar equations: Examples.- 20.3 Integrable scalar equations: Classification.- 20.4 Bibliographical remarks.- 21 Lie-algebraic Toda Systems.- 21.1 Introduction.- 21.2 Lie-algebraic open-end Toda lattices.- 21.3 Lie-algebraic periodic Toda lattices.- 21.4 Toda lattices AN-1and A 1.- 21.5 Discrete time lattices AN-1and 1.- 21.6 List of generalized Toda lattices.- 21.7 Discretization of lattices BN, CN, CN1), AzN, and DNZ+1.- 21.8 Discretization of lattices DN, D(Air), BN1, and A2N_1.- 21.9 2 x 2 Lax representations: continuous time case.- 21.10 2 x 2 Lax representations: discrete time case.- 21.11 Lattice G2as a reduction of the lattice B3.- 21.12 Lattice G as a reduction of the lattice B1).- 21.13 Toda lattice D(43), continuous and discrete.- 21.14 Bibliographical remarks.- 22 Gamier System.- 22.1 Introduction.- 22.2 Gamier system.- 22.2.1 Equations of motion and Hamiltonian structure.- 22.2.2 Integrals of motion.- 22.2.3 “Big” Lax representation.- 22.2.4 “Small” Lax representation.- 22.3 Anharmonic oscillator.- 22.3.1 Equations of motion and Hamiltonian structure.- 22.3.2 Integrals of motion.- 22.3.3 “Big” Lax representation.- 22.3.4 “Small” Lax representation.- 22.4 Wojciechowski system.- 22.4.1 Equations of motion and Hamiltonian structure.- 22.4.2 Integrals of motion.- 22.4.3 “Big” Lax representation.- 22.4.4 “Small” Lax representation.- 22.5 Bäcklund transformation for the Gamier system.- 22.6 Bäcklund transformation for anharmonic oscillator.- 22.7 Bäcklund transformation for Wojciechowski system.- 22.8 Explicit discretization of the Gamier system.- 22.8.1 Equations of motion and symplectic properties.- 22.8.2 Integrals of motion.- 22.8.3 “Big” Lax representation.- 22.8.4 “Small” Lax representation.- 22.9 Explicit discretization of anharmonic oscillator.- 22.9.1 Equations of motion and symplectic properties.- 22.9.2 Integrals of motion.- 22.9.3 “Big” Lax representation.- 22.9.4 “Small” Lax representation.- 22.10 Explicit discretization of Wojciechowski system.- 22.10.1 Equations of motion and symplectic properties.- 22.10.2 Integrals of motion.- 22.10.3 “Big” Lax representation.- 22.10.4 “Small” Lax representation.- 22.11 Bibliographical remarks.- 23 Hénon-Heiles System.- 23.1 Introduction.- 23.2 Lax representation.- 23.3 Discretization of Hénon-Heiles system.- 23.4 Bibliographical remarks.- 24 Neumann System.- 24.1 Introduction.- 24.2 Double Neumann system.- 24.2.1 Equations of motion and Hamiltonian structure.- 24.2.2 “Big” Lax representation.- 24.2.3 “Small” Lax representation.- 24.2.4 Unconstrained version.- 24.3 Neumann system.- 24.3.1 Equations of motion and Hamiltonian structure.- 24.3.2 “Big” Lax representation.- 24.3.3 “Small” Lax representation.- 24.3.4 Unconstrained version.- 24.4 Rosochatius system.- 24.4.1 Equations of motion and Hamiltonian structure.- 24.4.2 “Big” Lax representation.- 24.4.3 “Small” Lax representation.- 24.4.4 Unconstrained version.- 24.5 Bäcklund transformation for double Neumann system.- 24.6 Bäcklund transformation for Neumann system.- 24.7 Bäcklund transformation for Rosochatius system.- 24.8 Ragnisco’s discretization of Neumann system.- 24.9 V. Adler’s discretization of Neumann system.- 24.10 Coupled Neumann system.- 24.11 Discretizations of the coupled Neumann system.- 24.11.1 Discretization à la Ragnisco.- 24.11.2 Discretization à la V. Adler.- 24.12 Bibliographical remarks.- 25 Lie-algebraic Generalizations of the Gamier Systems.- 25.1 Introduction.- 25.2 Gamier systems related to symmetric spaces.- 25.3 Gamier systems related to AIII.- 25.3.1 Equations of motion and Lax representation.- 25.3.2 Example: M = 2.- 25.3.3 Discretizations.- 25.3.4 Example: discrete systems with M = 2.- 25.4 Gamier systems related to CI and DIII.- 25.5 Gamier systems related to BDI.- 25.5.1 Equations of motion and Lax representation.- 25.5.2 Discretization.- 25.6 Bibliographical remarks.- 26 Integrable Cases of Rigid Body Dynamics.- 26.1 Introduction.- 26.2 Multi-dimensional Euler top.- 26.3 Discrete time Euler top.- 26.4 Rigid body in a quadratic potential.- 26.5 Discrete time top in a quadratic potential.- 26.6 Multi-dimensional Lagrange top.- 26.6.1 Body frame formulation.- 26.6.2 Rest frame formulation.- 26.7 Discrete time analog of the Lagrange top.- 26.7.1 Rest frame formulation.- 26.7.2 Moving frame formulation.- 26.8 Three-dimensional Lagrange top.- 26.9 Discrete time three-dimensional Lagrange top.- 26.10 Rigid body motion in an ideal fluid: Clebsch case.- 26.11 Discretization of the Clebsch problem.- 26.11.1 Case A = B2of the Clebsch problem.- 26.11.2 Case A = B of the Clebsch problem.- A Appendix: Lagrange top and Heisenberg magnetic.- 26.12 Bibliographical remarks.- 27 Systems of Calogero-Moser Type.- 27.1 Introduction.- 27.2 Lax representations: rational and hyperbolic cases.- 27.3 Dynamical r-matrix formulation.- 27.4 Explicit solutions.- 27.4.1 Rational systems.- 27.4.2 Hyperbolic systems.- 27.5 Discrete time evolution: rational systems.- 27.5.1 Rational CM system.- 27.5.2 Rational RS system.- 27.6 Discrete time evolution: hyperbolic systems.- 27.6.1 Hyperbolic CM system.- 27.6.2 Hyperbolic RS systems.- 27.7 Elliptic CM type models: Lax representations.- 27.8 Elliptic CM type models: r-matrix structure.- 27.9 Discretization of elliptic CM and RS models.- 27.10 Strong coupling limit of RS models.- 27.10.1 Rational system.- 27.10.2 Hyperbolic system.- 27.11 Bibliographical remarks.- List of Notations.

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