Applications of Lie Groups to Differential Equations
Symmetry methods have long been recognized to be of great importance for the study of the differential equations. This book provides a solid introduction to those applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented so that graduate students and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter.
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according acting action appear apply basic characteristic coefficients complete computation condition connected conservation law Consider constant construction coordinates corresponding defined Definition dependent variables derivatives determined differential equations differential operator discussed divergence equivalent Euler-Lagrange equations evolution equations Example Exercise exist expression fact formal formula function further G-invariant given group-invariant solutions heat hence holds identity important independent infinitesimal integral invariant leading Lie algebra Lie group linear manifold matrix method Note one-parameter operator orbits ordinary differential equations original particular Poisson bracket prolongation proof properties Proposition Prove rank reduced respect result rotations satisfy smooth function solutions solve space structure subgroup submanifold subset Suppose symmetry group system of differential takes Theorem theory tion translations trivial vanishes variables variational symmetry vector field wave ди