Constrained Minimisation Problems (Method of Lagrange Multipliers)

Part of the Texts and Monographs in Physics book series (TMP)


In many applications of the calculus of variations, what we want to determine is not just the minimum of a function f on an open set U but the minimum of f subject to certain restrictions on the points xU. A well-known example from classical mechanics can be used to illustrate this type of problem. We want to determine the minimum of the action functional subject to the subsidiary condition that the motion be on a given surface. The restriction in this case is, therefore, that the points xU. satisfy an equation of the form g(x) = 0, i.e. the equation of the surface.


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  1. 4.1
    Ljusternik, L.A.: On conditional extrema of functions. Mat. Sbornik 41, 3 (1934)Google Scholar
  2. 4.2
    Robertson, A.P., Robertson, W.J.: Topological vector spaces. Cambridge University Press, Cambridge 1973zbMATHGoogle Scholar
  3. 4.3
    Dieudonné, J.: Foundations of modern analysis. Academic Press, New York 1969zbMATHGoogle Scholar
  4. 4.4
    Hirzebruch, F., Scharlau, W.: Einführung in die Funktionalanalysis. BI Taschenbücher 296, BI, Mannheim 1971zbMATHGoogle Scholar
  5. 4.5
    Gelfand, I.M., Fomin, S.V.: Calculus of variations. Prentice-Hall, Englewood Cliffs 1963Google Scholar

Further Reading

  1. Schwartz, J.T.: Nonlinear functional analysis. Gordon and Breach, New York 1969zbMATHGoogle Scholar
  2. Maurin, K.: Calculus of variations and classical field theory, Part 1. Lecture Notes Series 34, Matematisk Institut, Aarhus University 1976Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Theoretische Physik, Fakultät für PhysikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa

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