This book is devoted to explaining a wide range of applications of continuous symmetry groups to physically important systems of differential
equations. Emphasis is placed on significant applications of group-theoretic
methods, organized so that the applied reader can readily learn the basic
computational techniques required for genuine physical problems. The first
chapter collects together (but does not prove) those aspects of Lie group
theory which are of importance to differential equations. Applications covered
in the body of the book include calculation of symmetry groups of differential
equations, integration of ordinary differential equations, including special
techniques for Euler-Lagrange equations or Hamiltonian systems,
differential invariants and construction of equations with prescribed symmetry
groups, group-invariant solutions of partial differential equations,
dimensional analysis, and the connections between conservation laws and
symmetry groups. Generalizations of the basic symmetry group concept, and
applications to conservation laws, integrability conditions, completely
integrate systems and soliton equations, and bi-Hamiltonian systems are covered
in detail. The exposition is reasonably self-contained, and supplemented by
numerous examples of direct physical importance, chosen from classical
mechanics, fluid mechanics, elasticity and other applied areas. Besides the basic
theory of manifolds. Lie groups and algebras, transformation groups and
differential forms, the book delves into the more theoretical subjects of
prolongation theory and differential equations, the Cauchy-Kovalevskaya
theorem, characteristics and integrability of differential equations, extended jet
spaces over manifolds, quotient manifolds, adjoint and co-adjoint
representations of Lie groups, the calculus of variations and the inverse problem of
characterizing those systems which are Euler-Lagrange equations of some
variational problem, differential operators, higher Euler operators and the
variational complex, and the general theory of Poisson structures, both for
finite-dimensional Hamiltonian systems as well as systems of evolution
equations, all of which have direct bearing on the symmetry analysis of differential
equations. It is hoped that after reading this book, the reader will, with
a minimum of difficulty, be able to readily apply these important group-
theoretic methods to the systems of differential equations he or she is
interested in, and make new and interesting deductions concerning them. If so, the
book can be said to have served its purpose.