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)


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 \(f_{{x^\alpha }}^i\left( {{x^\beta }} \right) = \left( {\partial /\partial {x^\alpha }} \right){f^i}\left( {{x^\beta }} \right)\).


Mercury Univer Hone Bonnet Nexion 


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Copyright information

© Springer Science+Business Media New York 1984

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

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