Abstract
Gradient extremals on N-dimensional energy hypersurfaces V=V(x 1 ⋯ x n ) are curves defined by the condition that the gradient ∇V is an eigenvector of the hessian matrix ∇∇V. For variations which are restricted to any (N−1) dimensional hypersurface ∇V(x 1⋯ x N ) = V 0= constant, the absolute value of the gradient ∇V is an extremum at those points where a gradient extremal intersects this surface. In many, though not all, cases gradient extremals go along the bottom of a valley or along the crest of a ridge. The properties of gradient extremals are discussed through a detailed differential analysis and illustrated by an explicit example. Multidimensional generalizations of gradient extremals are defined and discussed.
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Similar results were also reported by Müller, K (1980) Angew Chem Int Ed Engl 19:1
Pancir J (1975) Collect Czech Chem Commun 40:1112
Basilevsky MV, Shamov AG (1981) Chem Phys 60:347
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Operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences
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Hoffman, D.K., Nord, R.S. & Ruedenberg, K. Gradient extremals. Theoret. Chim. Acta 69, 265–279 (1986). https://doi.org/10.1007/BF00527704
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DOI: https://doi.org/10.1007/BF00527704