# The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants

• B. A. Dubrovin
• A. T. Fomenko
• S. P. Novikov
Part of the Graduate Texts in Mathematics book series (GTM, volume 93)

## Abstract

Let OD denote a region with piecewise smooth boundary D, of the Euclidean space I1’ with Euclidean co-ordinates x1,..., xn. Consider the linear space F of smooth vector-functions f (x1,..., xn) = (f1,..., fk) defined on D, i.e. with domain D. Let L(x β;p j ;q a i )be a smooth real-valued function of the three arguments xβ, 1 ≤β ≤ n; p j , 1jk; q a i , 1ik, 1an (making altogether n + k + nk real arguments); we call such a function a Lagrangian, and from a given such Lagrangian we construct a functional I[f] defined on F, as follows:
$$I\left[ f \right] = \int_D {L\left( {{x^\beta };{f^j}\left( {{x^\beta }} \right);f_{{x^\alpha }}^i\left( {{x^\beta }} \right)} \right)} d{x^1} \wedge \cdots \wedge d{x^n}$$
where the integral is the multiple integral (see §26) over the region D (which we shall later assume to be bounded), and where $$f_{{x^\alpha }}^i\left( {{x^\beta }} \right) = \left( {\partial /\partial {x^\alpha }} \right){f^i}\left( {{x^\beta }} \right)$$.

### Keywords

Mercury Univer Hone Bonnet Nexion

## Authors and Affiliations

• B. A. Dubrovin
• 1
• A. T. Fomenko
• 2
• S. P. Novikov
• 3
1. 1.c/o VAAP-Copyright Agency of the U.S.S.R.MoscowUSSR
2. 2.3 Ya KaracharavskayaMoscowUSSR
3. 3.L. D. Landau Institute for Theoretical PhysicsAcademy of Sciences of the U.S.S.R.MoscowUSSR