Multi-Hamiltonian Theory of Dynamical Systems

1. Preliminary Considerations.- 2. Elements of Differential Calculus for Tensor Fields.- 2.1 Tensors.- 2.2 Tensor Fields.- 2.3 Transformation Properties of Tensor Fields.- 2.4 Directional Derivative of Tensor Fields.- 2.5 Differential ?-Forms.- 2.6 Flows and Lie Transport.- 2.7 Lie Derivatives.- 3. The Theory of Hamiltonian and Bi-Hamiltonian Systems.- 3.1 Lie Algebras.- 3.2 Hamiltonian and Bi-Hamiltonian Vector Fields.- 3.3 Symmetries and Conserved Quantities of Dynamical Systems.- 3.4 Tensor Invariants of Dynamical Systems.- 3.5 Algebraic Properties of Tensor Invariants.- 3.6 The Miura Transformation.- 4. Lax Representations of Multi-Hamiltonian Systems.- 4.1 Lax Operators and Their Spectral Deformations.- 4.2 Lax Representations of Isospectral and Nonisospectral Hierarchies.- 4.3 The Lax Operator Algebra.- 5. Soliton Particles.- 5.1 General Aspects.- 5.2 Algebraic Structure of Linear Systems.- 5.3 Algebraic Structure of Multi-Soliton Representation.- 5.4 Multi-Soliton Perturbation Theory.- 6. Multi-Hamiltonian Finite Dimensional Systems.- 6.1 Stationary Flows of Infinite Systems. Ostrogradsky Parametrizations.- 6.2 Stationary Flows of Infinite Systems. Newton Parametrization.- 6.3 Constrained Flows of Lax Equations.- 6.4 Restricted Flows of Infinite Systems.- 6.5 Separability of Bi-Hamiltonian Chains with Degenerate Poisson Structures.- 6.6 Nonstandard Multi-Hamiltonian Structures and Their Finite Dimensional Reductions.- 6.7 Bi-Hamiltonian Chains on Poisson-Nijenhuis Manifolds.- 7. Multi-Hamiltonian Lax Dynamics in (1+1)-Dimensions.- 7.1 Hamiltonian Dynamics on Lie Algebras.- 7.2 Basic Facts About R-Structures.- 7.3 Multi-Hamiltonian Dynamics of Pseudo-Differential Lax Operators.- 7.4 Multi-Hamiltonian Dynamics of Shift Lax Operators.- 8. Towards a Multi-Hamiltonian Theory of (2+1)-Dimensional Field Systems.- 8.1 The Sato Theory.- 8.2 Multi-Hamiltonian Lax Dynamics for Noncommutative Variables.- References.