Equivalence, Invariants and Symmetry
This book presents an innovative synthesis of methods used to study the problems of equivalence and symmetry that arise in a variety of mathematical fields and physical applications. It draws on a wide range of disciplines, including geometry, analysis, applied mathematics, and algebra. Dr. Olver develops systematic and constructive methods for solving equivalence problems and calculating symmetries, and applies them to a variety of mathematical systems, including differential equations, variational problems, manifolds, Riemannian metrics, polynomials, and differential operators. He emphasizes the construction and classification of invariants and reductions of complicated objects to simple canonical forms. This book will be a valuable resource for students and researchers in geometry, analysis, algebra, mathematical physics and related fields.
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Jets and Contact Transformations
Symmetries of Differential Equations
Cartans Equivalence Method
Prolongation of Equivalence Problems
Anderson R L 119 129 11
The CartanKahler Existence Theorem
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associated canonical form Cartan form change of variables classifying manifolds coframe derivatives compute constant contact form contact transformation contact-invariant covariants defined Definition denote dependent variables derived invariants differential forms differential operators differential system equivalence problem essential torsion coefficients Euler-Lagrange equation Example Exercise fiber-preserving finite-dimensional formula G-invariant given GL(n group action group parameters hence higher order implies infinitesimal integral element integral submanifold invariant coframe invariant differential involutive jet space Lagrangian Lie algebra Lie bracket lifted coframe linear matrix Maurer-Cartan forms multiplier representation normalize one-forms open subset orbit dimension order derivative order differential invariants order ordinary differential ordinary differential equation partial differential equations point transformation polynomial prolonged proof Proposition Prove rank regular relative invariant satisfy scalar smooth solution structure equations structure functions structure group structure invariants subgroup symmetry group tangent vectors Theorem tion torsion coefficients transformation group two-forms vanish variational problem vector field vector field system