Hamilton Jacobi Theory in the Calculus of Variations Its Role in Mathematics Theory and Application

by Hanno Rund

"This rigorous self-contained account demonstrates the role of the calculus of variations in unifying some of the most fundamental branches of pure mathematics and theoretical physics. The importance of the dHamilton Jacobi theory is stressed from the start, and so the pure mathematician gains immediate access to the theory of first-order partial differential equations, to that of some second-order partial differential equations, and to metric geometries. The theoretical physicist is shown how the theory of non=homogeneous single integral problems give rise to relativistic particle mechanics, in which the special invariant Hamiltonian function permits a particularly simple method of quantization, from which the relativistic wave equations C(Dirac, Kemmer, etc.) may be obtained directly. A very substantial part is devoted to multiple integral problems, with special reference to modern field theories and areal spaces. The same fundamental methods are used to discuss rigorously the problem of Lagrange, and they therefore lead to an analysis of fundamental dynamical laws for both holonomic and non-holonomic systems. The fundamental pure mathematics is derived from Caratheodory's approach to the calculus of variations, of which no account exists in English, although his methods may be said to have revolutionized the most basic aspects of the subjects. Many of the methods and some of the results now presented are original, and their is little overlap with existing literature, particularly on multiple integral theories".

452 pages, Hardcover