Applications of Lie Groups to Differential Equations

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.