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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 7. Legendre Transformation, Hamiltonian Systems, Convexity, Field Theories

Abstract
This chapter links the first half of our treatise to the second by preparing the transition from the Euler—Largrange formalism of the calculus of variations to the canonical formalism of Hamilton—Jacobi, which in some sense is the dual picture of the first. The duality transformation transforming one formalism into the other is the so-called Legendre transformation derived from the Lagrangian F of the variational problem that we are to consider. This transformation yields a global diffeomorphism and is therefore particularly powerful if F(x, z, p) is elliptic (i.e. uniformly convex) with respect to p. Thus the central themes of this chapter are duality and convexity.
Mariano Giaquinta, Stefan Hildebrandt

### Chapter 8. Parametric Variational Integrals

Abstract
In this chapter we shall treat the theory of one-dimensional variational problems in parametric form. Problems of this kind are concerned with integrals of the form
$$F(x) = \int_a^b {F(x(t))} ,\dot x(t))dt$$
(1)
, whose integrand F(x, υ)is positively homogeneous of first degree with respect to υ. Such integrals are invariant with respect to transformations of the parameter t, and therefore they play an important role in geometry. A very important example of integrals of the type (1) is furnished by the weighted arc length
$$S(x): = \int_a^b {\omega (x(t))} \left| {\dot x(t)} \right|dt$$
(2)
, which has the Lagrangian F(x, υ) = ω(x)|υ|. Many celebrated questions in differential geometry and mechanics lead to variational problems for parametric integrals of the form (2), and because of Fermat’s principle also the theory of light rays in isotropic media is governed by the integral (2), whereas the geometrical optics of general anisotropic media is just the theory of extremals of the integral (1).
Mariano Giaquinta, Stefan Hildebrandt

### Chapter 9. Hamilton-Jacobi Theory and Canonical Transformations

Abstract
In this chapter we want to present the basic features of the Hamilton—Jacobi theory, the centerpiece of analytical mechanics, which has played a major role in the development of the mathematical foundations of quantum mechanics as well as in the genesis of an analysis on manifolds. This theory is not only based on the fundamental work of Hamilton and Jacobi, but it also incorporates ideas of predecessors such as Fermat, Newton, Huygens and Johann Bernoulli among the old masters and Euler, Lagrange, Legendre, Monge, Pfaff, Poisson and Cauchy of the next generations. In addition the contributions of Lie, Poincaré and E. Cartan had a great influence on its final shaping.
Mariano Giaquinta, Stefan Hildebrandt

### Chapter 10. Partial Differential Equations of First Order and Contact Transformations

Abstract
This chapter can to a large extent be read independently of the others and serves as an introduction to the theory of partial differential equations of first order and to Lie’s theory of contact transformations. Nevertheless the results presented here are closely related to the rest of the book, in particular to field theory (Chapter 6) and to Hamilton—Jacobi theory (Chapter 9).
Mariano Giaquinta, Stefan Hildebrandt

### Backmatter

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