The Formulation of Variational Principles by Means of Clebsch Potentials

  • Hanno Rund
Part of the Studies in the Natural Sciences book series (SNS, volume 13)


Although Clebsch representations have appeared sporadically in the literature on classical hydrodynamics, the applicability of such representations to physics in general, and variational principles in particular, seems to have been overlooked until fairly recently. This is probably due to the fact that the standard formulation of such representations, originally 2 due to A. Clebsch, is subject to severe restri citions, as is immediately evident from the following enunciation: Given any differentiable vector field X on a 3-dimensional Euclidean space E3, there exist scalar functions Ψ, P, Q of the coordinates of E3, such that X can be represented in the form
$$ \vec X = \nabla \Psi + Q\nabla P $$
The aforementioned restricitions refer to the dimension of the space, as well as to its Euclidean character. Thus, when applications of a general nature are envisaged, one is compelled to seek extensions (if such are feasible) which are not subject to the limitations inherent in (1.1).


Variational Principle Lagrange Density Magnetic Charge Canonical Equation Differentiable Vector Field 
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Copyright information

© Plenum Press, New York 1977

Authors and Affiliations

  • Hanno Rund
    • 1
  1. 1.Department of MathematicsThe University of ArizonaTucsonUSA

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