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 weIl as Hamilton- Jacobi theory and the
classical theory of partial differential equations of first order. 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 monotonicity 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 non para- metric and parametric integrals and its relations
to Hamilton-Jacobi theory, geometrieal 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
several instances.
