Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie)

CONTENTSPrefaceChapter I. Definitions and elementary applications §1.1. One-parameter transformation groups §1.2. Prolongation formulae §1.3. Groups admissible by differential equations §1.4. Integration and reduction of order using one-parameter groups1.4.1. Integrating factor1.4.2. Method of canonical variables1.4.3. Invariant differentiation §1.5. Defining equations §1.6. Lie algebras §1.7. Contact transformationsChapter II. Integration of second-order equations admitting a two-dimensional algebra §2.1. Consecutive reduction of order2.1.1. An instructive example2.1.2. Solvable Lie algebras §2.2. The method of canonical variables2.2.1. Changes of variables and basis in an algebra2.2.2. Canonical form of two-dimensional algebras2.2.3. An integration algorithm2.2.4. An example of implementation of the algorithmChapter III. Group-theoretical classification of second-order equations §3.1. Equations admitting a three-dimensional algebra3.1.1. Classification in the complex domain3.1.2. Classification over the reals. Isomorphism and similarity §3.2. The general classification result §3.3. Two remarkable classes of equations3.3.1. The equation Linearizability criteria3.3.2. Equations Chapter IV. Ordinary differential equations with a fundamental system of solutions (following Vessiot-Guldberg-Lie) §4.1. The main theorem §4.2. Examples §4.3. Projective interpretation of the Riccati equation §4.4. Linearizable Riccati equationsChapter V. The invariance principle in problems of mathematical physics §5.1. Spherical functions §5.2. A group-theoretical touch to Riemann's method §5.3. Symmetry of fundamental solutions, or the first steps in group analysis in the space of distributions5.3.1. Something about distributions5.3.2. Laplace's equation5.3.3. The heat equation5.3.4. The wave equationChapter VI. Summary of resultsReferences