## Multi-Hamiltonian Theory of Dynamical SystemsA modern Hamiltonian theory offering a unified treatment of all types of systems (i.e. finite, lattice, and field) is presented. Particular attention is paid to nonlinear systems that have more than one Hamiltonian formulation in a single coordinate system. As this property is closely related to integrability, this book presents an algebraic theory of integrable systems. The book is intended for scientists, lecturers, and students interested in the field. |

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### Contents

Preliminary Considerations | 1 |

Elements of Differential Calculus for Tensor Fields | 13 |

The Theory of Hamiltonian and BiHamiltonian Systems | 41 |

Copyright | |

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### Common terms and phrases

1)-dimensional adjoint AKNS arbitrary bi-Hamiltonian chain bi-Hamiltonian formulation canonical Casimir commutator compatible conserved quantities const constraint construct cosymmetries covector defined denote differential operator dynamical system eigenfunctions equivalent evolution equations Example field systems finite dimensional given Hamiltonian formulation Hamiltonian representation Hamiltonian system Hamiltonian vector field Hence hereditary algebra implectic operator infinite integrable inverse isospectral Jacobi identity KdV equation KdV hierarchy Lagrangian lattice Lax equations Lax operators Lax pair Lax representation Lemma Lie algebra Lie derivative linear manifold master symmetries matrix Miura map MKdV modified multi-Hamiltonian N-soliton Newton representation Nijenhuis coordinates nonisospectral nonlinear obtain Ostrogradsky representation pair parameter phase space Poisson bracket Poisson operator Poisson structure Poisson tensor Proof recursion operator respect restricted flows satisfies scalar fields soliton solutions spectral problem stationary flow subalgebra subspace symplectic takes the form tensor field tensor invariants Theorem theory transformation zero-curvature