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/content/aip/journal/jmp/19/8/10.1063/1.523879
1.
1.H. F. Ahner and A. E. Moose, J. Math Phys. 18, 1367 (1977).
2.
2.R. W. Atherton and G. M. Homsy, Stud. Appl. Math. 54, 31 (1975).
3.
3.We appreciate communications with D. Lovelock, R. Pavelle, I. Anderson, G. Horndeski, and S. Aldersley. They pointed out an error in Ref. 1. The word “equation” should be replaced by the words “scalar equation” in the sentence following Eq. (24). For a tensor equation the potential conditions may be satisfied when the field operators are of odd order. A considerable body of work exists that we were unaware of in which results similar to ours were derived by other methods. Some of the most pertinent references include: D. Lovelock and H. Rund, Tensors, Differential Forms and Variational Principles (Wiley, New York, 1975), Chap. 8 and references therein;
3.G. W. Horndeski, Tensor 28, 309 (1974);
3.G. W. Horndeski, 29, 21 (1975);
3.G. W. Horndeski, J. Math. Phys. 17, 1980 (1976);
3.D. Lovelock, J. Math. Phys. 18, 1491 (1977);
3.I. M. Anderson, “Tensorial Euler‐Lagrange Expressions and Conservation Laws” to appear in Aequations Mathematicae. For flat space considerations see also R. M. Santilli, Ann. Phys. N.Y. 103, 354 (1977);
3.I. M. Anderson, 103, 409 (1977);
3.I. M. Anderson, 105, 227 (1977).
3.For treatment of antiderivatives, of variational principles that yield integral equations, and of definitional questions, see E. P. Hamilton (to appear in J. Math. Anal. Appl.);
3.E. P. Hamilton and B. E. Goodwin, in Analytic Methods in Mathematical Physics, edited by R. P. Gilbert and R. G. Newton (Gordon and Breach, New York, 1970);
3.E. P. Hamilton, “A New Definition of Variational Derivative” (preprint).
4.
4.Expression (1) fails also when (1) is identically zero.
5.
5.Some cases are considered in S. J. Aldersley, “Higher Euler Operators and Some of their Applications” (to appear in J. Math. Phys.).
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2008-08-11
2016-12-09
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