1 Introduction to Lie Groups. - 1. 1. Manifolds. - 1. 2. Lie Groups. - 1. 3. Vector Fields. - 1. 4. Lie Algebras. - 1. 5. Differential Forms. - Notes. - Exercises. - 2 Symmetry Groups of Differential Equations. - 2. 1. Symmetries of Algebraic Equations. - 2. 2. Groups and Differential Equations. - 2. 3. Prolongation. - 2. 4. Calculation of Symmetry Groups. - 2. 5. Integration of Ordinary Differential Equations. - 2. 6. Nondegeneracy Conditions for Differential Equations. - Notes. - Exercises. - 3 Group-Invariant Solutions. - 3. 1. Construction of Group-Invariant Solutions. - 3. 2. Examples of Group-Invariant Solutions. - 3. 3. Classification of Group-Invariant Solutions. - 3. 4. Quotient Manifolds. - 3. 5. Group-Invariant Prolongations and Reduction. - Notes. - Exercises. - 4 Symmetry Groups and Conservation Laws. - 4. 1. The Calculus of Variations. - 4. 2. Variational Symmetries. - 4. 3. Conservation Laws. - 4. 4. Noether's Theorem. - Notes. - Exercises. - 5 Generalized Symmetries. - 5. 1. Generalized Symmetries of Differential Equations. - 5. 2. Récursion Operators Master Symmetries and Formal Symmetries. - 5. 3. Generalized Symmetries and Conservation Laws. - 5. 4. The Variational Complex. - Notes. - Exercises. - 6 Finite-Dimensional Hamiltonian Systems. - 6. 1. Poisson Brackets. - 6. 2. Symplectic Structures and Foliations. - 6. 3. Symmetries First Integrals and Reduction of Order. - Notes. - Exercises. - 7 Hamiltonian Methods for Evolution Equations. - 7. 1. Poisson Brackets. - 7. 2. Symmetries and Conservation Laws. - 7. 3. Bi-Hamiltonian Systems. - Notes. - Exercises. - References. - Symbol Index. - Author Index. Language: English