In Exterior Differential Systems, the authors present the results of their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincaré-Cartan forms. They also cover certain aspects of the theory of exterior differential systems, which provides the language and techniques for the entire study. Because it plays a central role in uncovering geometric properties of differential equations, the method of equivalence is particularly emphasized. In addition, the authors discuss conformally invariant systems at length, including results on the classification and application of symmetries and conservation laws. The book also covers the Second Variation, Euler-Lagrange PDE systems, and higher-order conservation laws.This timely synthesis of partial differential equations and differential geometry will be of fundamental importance to both students and experienced researchers working in geometric analysis.

Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Chicago Lectures in Mathematics)

Science & Math

Daniel Grossman

Phillip Griffiths

Robert L. Bryant

Mathematics

University of Chicago Press

This book gives a treatment of exterior differential systems. It will in clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of inde pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepen dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object.

Exterior Differential Systems (Mathematical Sciences Research Institute Publications)

Springer

<p>In the period since the original four volumes of Phillip Griffiths's Selecta were published (Selected Works of Phillip A. Griffiths with Commentary), Parts 1-4, Collected Works, Volume 18), Griffiths has continued to produce beautiful and important work. The current two-part publication brings Griffiths's Selecta up to date by including the majority of his recent articles, as well as two older papers on differential geometry whose length had precluded their inclusion in the original Selecta.The papers are organized along the three main topics: Differential Geometry and Hodge Theory (Part 5) and Algebraic Cycles (Part 6). In addition to his papers, Griffiths has been an author of a number of research monographs. To give the reader an overview of what these monographs contain, introductions to some of these are also included.</p>

Selected Works of Philip A. Griffiths With Commentary (Collected Works) (Collected Works, 26)

American Mathematical Society

Selected Works of Philip A. Griffiths With Commentary: Differential Geometry and Hodge Theory (Collected Works) (Collected Works, 26)

Selected Works of Philip A. Griffiths With Commentary: Algebraic Cycles (Collected Works) (Collected Works, 26)

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