This book is devoted to the development of geometrie methods for studying and revealing geometrie aspects of the theory of differential equations with quadratie right-hand sides (Riccati-type equations), which are closely related to the calculus of variations and optimal control theory. The book contains the following three parts, to each of which aseparate book could be devoted: 1. the classieal calculus of variations and the geometrie theory of the Riccati equation (Chaps. 1-5), 2. complex Riccati equations as flows on Cartan-Siegel homogeneity da mains (Chap. 6), and 3. the minimization problem for multiple integrals and Riccati partial dif ferential equations (Chaps. 7 and 8). Chapters 1-4 are mainly auxiliary. To make the presentation complete and self-contained, I here review the standard facts (needed in what folIows) from the calculus of variations, Lie groups and algebras, and the geometry of Grass mann and Lagrange-Grassmann manifolds. When choosing these facts, I pre fer to present not the most general but the simplest assertions. Moreover, I try to organize the presentation so that it is not obscured by formal and technical details and, at the same time, is sufficiently precise. Other chapters contain my results concerning the matrix double ratio, com plex Riccati equations, and also the Riccati partial differential equation, whieh the minimization problem for a multiple integral. arises in The book is based on a course of lectures given in the Department of Me and Mathematics of Moscow State University during several years.

The only monograph on this topic

Inhalt

1. Classical Calculus of Variations.- 2. Riccati Equation in the Classical Calculus of Variations.- 3. Lie Groups and Lie Algebras.- 4. Grassmann Manifolds.- 5. Matrix Double Ratio.- 6. Complex Riccati Equations.- 7. Higher-Dimensional Calculus of Variations.- 8. On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral.- Epilogue.- Appendix to the English Edition.- References.