Contents -- Preface -- Introduction -- I Analytical Mechanics -- 1 The Lagrangian Coordinates -- 1.1 A Primer for Various Formulations of Dynamics -- 1.2 Constraints -- 1.3 Degrees of Freedom and Lagrangian Coordinates -- 1.4 The Calculus of Variations and the Lagrange Equations -- 1.5 Remarks on Lagrange's Equations -- 2 Hamiltonian Systems -- 2.1 The Legendre Transformation -- 2.2 The Hamilton Equations -- 2.3 The Poisson Bracket and the Jacobi-Poisson Theorem -- 2.4 A More Compact Form of The Hamiltonian Dynamics -- 2.5 The Variational Principle for the Hamilton Equations -- 3 Transformation Theory -- 3.1 Canonical, Completely Canonical and Symplectic Transformations -- 3.2 A New Characterization of Completely Canonical Transformations -- 3.3 New Characterization of Sympletic Transformations -- 4 The Integration Methods -- 4.1 Integrals Invariants of a Differential System -- 4.2 A Primer on the Lie Derivative -- 4.3 The Kepler Dynamics -- 4.4 The Hamilton-Jacobi Integration Method -- 4.5 The Hamilton-Jacobi Equation for the Kepler Potential -- 4.6 The Liouville Theorem on the Complete Integrability -- II Basic Ideas of Differential Geometry -- 5 Manifolds and Tangent Spaces -- 5.1 Differential Manifolds -- 5.2 Curves on a Differential Manifold -- 5.3 Tangent Space at a Point -- 5.4 A Digression on Vectors and Covectors -- 5.5 Cotangent Space at a Point -- 5.6 Maps Between Manifolds -- 5.7 Vector Fields -- 5.8 The Tangent Bundle -- 5.9 General Definition of Fiber Bundle -- 5.10 Integral Curves of a Vector Field -- 5.11 The Lie Derivative -- 5.12 Submanifolds -- 6 Differential Forms -- 6.1 The Tensors -- 6.2 The Tensor Fields -- 6.3 The Metric Tensor Field on a Manifold -- 6.4 Endomorphisms Associated with a Mixed Tensor Field -- 7 Integration Theory -- 7.1 Orientable Manifolds.

7.2 Integration on Orientable Manifolds -- 7.3 p-Vectors and Dual Tensors -- 7.4 Metric o Volume = Hodge Duality -- 7.5 Stokes Theorem -- 7.6 Gradient, Curl and Divergence -- 7.7 A Primer for Cohomology -- 7.8 Scalar Product of Differential p-Forms -- 8 Lie Groups and Lie Algebras -- 8.1 Lie Groups -- 8.2 Building of a Lie Algebra from a Lie Group -- III Geometry and Physics -- 9 Symplectic Manifolds and Hamiltonian Systems -- 9.1 Symplectic Structures on a Manifold -- 9.2 Locally and Globally Hamiltonian Vector Fields -- 9.3 Hamiltonian Flows -- 9.4 The Cotangent Bundle and Its Symplectic Structure -- 9.5 Revisited Analytical Mechanics -- 9.6 The Liouville Theorem -- 9.7 A New Characterization of Complete Integrability -- 9.8 Applications -- 9.9 Poisson-Nijenhuis Structures -- 10 The Orbits Method -- 10.1 Reduced Phase Space -- 10.2 Orbits of a Lie Group in the Coadjoint Representation -- 10.3 The Rigid Body -- 10.4 Rigid Body Equations -- 11 Classical Electrodynamics -- 11.1 Maxwell's Equations -- 11.2 Geometrical Identification of Fields on R³ -- 11.3 Geometrical Identification of Electromagnetic Field in Space-Time -- IV Integrable Field Theories -- 12 KdV Equation -- 12.1 An Existence and Uniqueness Theorem -- 12.2 Symmetries -- 12.3 Conservation Laws -- 12.4 KdV as a Hamiltonian Dynamics -- 12.5 KdV as a Completely Integrable Hamiltonian Dynamics -- 13 General Structures -- 13.1 Notation and Generalities -- 13.2 Strongly and Weakly Symplectic Forms -- 13.3 Invariant Endomorphism -- 13.4 Invariant Endomorphisms and Liouville's Integrability -- 13.5 Recursion Operators in Dissipative Dynamics -- 14 Meaning and Existence of Recursion Operators -- 14.1 Integrable Systems -- 15 Miscellanea -- 15.1 A Tensorial Version of the Lax Representation -- 15.2 Liouville Integrability of Schrödinger Equation.

15.3 Integrable Systems on Lie Group Coadjoint Orbits -- 15.4 Deformation of a Lie Algebra -- 16 Integrability of Fermionic Dynamics -- 16.1 Recursion Operators in the Bosonic Case -- 16.2 Graded Differential Calculus -- 16.3 Poisson Supermanifold -- A Lagrange: A Short Biography -- B Concerning the Lie Derivative -- C Concerning the Kepler Action Variables -- D Concerning the Reduced Phase Space -- E On the Canonical Differential 1-Form -- F Concerning Rigid Body Equations -- G The Gelfand-Levitan-Marchenko Equation -- Bibliography -- Index.