This book describes the classical aspects of the variational calculus
which are of interest to analysts, geometers and physicists alike.
Volume 1 deals with the for- mal apparatus of the variational calculus
and with nonparametric field theory, whereas Volume 2 treats parametric
variational problems as well as Hamilton- Jacobi theory and the
classical theory of partial differential equations of first ordel;. In a
subsequent treatise we shall describe developments arising from
Hilbert's 19th and 20th problems, especially direct methods and
regularity theory. Of the classical variational calculus we have
particularly emphasized the often neglected theory of inner variations,
i. e. of variations of the independent variables, which is a source of
useful information such as mono tonicity for- mulas, conformality
relations and conservation laws. The combined variation of dependent and
independent variables leads to the general conservation laws of Emmy
Noether, an important tool in exploiting symmetries. Other parts of this
volume deal with Legendre-Jacobi theory and with field theories. In
particular we give a detailed presentation of one-dimensional field
theory for nonpara- metric and parametric integrals and its relations to
Hamilton-Jacobi theory, geometrical optics and point mechanics. Moreover
we discuss various ways of exploiting the notion of convexity in the
calculus of variations, and field theory is certainly the most subtle
method to make use of convexity. We also stress the usefulness of the
concept of a null Lagrangian which plays an important role in we give an
exposition of Hamilton-Jacobi several instances.
