Syllabus
In this course, students will learn
-how to find point (generalized) symmetries of a given differential equation (of a system of differential equations),
-how to use these symmetries (which form a Lie group) to simplify or to solve given differential equations,
-how to find conservation laws (integrals of motion) using point (generalized) symmetries of Euler-Lagrange differential equations which are also symmetries of a corresponding variational functional,
-how to find a general form of (linear or nonlinear) differential equations of a given order which are invariant under a given Lie group of symmetries.
Basic notions of the theory of Lie groups of transformations which students will learn during lectures:
-one-parameter and r-parameter Lie group of point transformations,
-infinitesimal transformations and generators of point transformations,
-Lie theorems, Lie algebra of a Lie group of transformations, solvable Lie algebra,
-prolongations of point transformations and their infinitesimal generators,
-symmetry group of differential equations and infinitesimal criterion of invariance of differential equations,
-canonical coordinates and their use to reduce, or to solve differential equations,
-differential invariants and their use to reduce differential equations,
-generalized symmetries of differential equations,
-invariant solutions of differential equations, reduction of the number of variables,
-variational symmetry, infinitesimal criterion of invariance,
-general conservation laws and their characteristics, Noether theorem for point and generalized symmetries.
Annotation
Symmetries of equations of mathematical physics and their solution using these symmetries. General differential equations of a given symmetry.
General conservation laws for systems of differential equations and their relation to symmetries of these equations.