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Generalized Hamilton–Jacobi theories
1.C. Carathéodory, Variationsrechnung und partielle Differential‐Gleichungen erster Ordnung (Teubner, Leipzig and Berlin, 1935;
1.translation, Holden‐Day, San Francisco, 1967), Sec. 239.
2.H. Rund, Arch. Ration. Mech. Anal. 65, 305 (1977).
3.Latin indices shall run from 1 to n; summation over repeated suffixes is implied.
4.Ref. 1. Sec. 140.
5.Ref. 2, Sec. 4.
6.H. Rund, The Hamilton‐Jacobi Theory in the Calculus of Variations (Van Nostrand, New York, 1966;
6.Krieger reprint, New York, 1973), Ch. 3.
7.Most of the results in this paper are contained in: R. Baumeister, “Applications of Clebsch Potentials to Variational Principles in the Theory of Physical Fields,” Ph.D. thesis, The University of Arizona, 1977.
8.The “character” of is identical with the “Pfaffian class” of the 1‐form
9.Greek indices shall run from 1 to m, summation over repeated indices being implied.
10.A. Clebsch, J. reine angew. Math 56, 1 (1859).
10.(In this paper, Clebsch also obtains representations for fields with character n.).
11.Ref. 6, Ch. 2.
12.E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (Wiley, New York, 1974), pp. 30–66.
13.Ref. 1, Sec. 21.
14.Ref. 12, pp. 41–44; also H. Rund, Tydskrif vir Natuurwetenskappe, June‐Sept. 1969, pp. 141–90.
15.This fact is also noted by H. Rund in Ref. 2, p. 319.
16.It should be stressed that Φ vanishes only for a special gauge, and that the results obtained when are gauge dependent.
17.Ref. 6, Ch. 3.
18.It follows that H is not to be identified with the energy of our dynamical system.
19.An analogous “associated multiple integral variational principle” for non‐relativistic dynamical systems may be found in Ref. 2, pp. 319–23.
20.Ref. 10, p. 9.
21.This result is a special case of a theorem established by H. Rund (personal communication). A proof may be found in Ref. 7, pp. 134–40.
22.This example is generalized in variational principles formulated by the author (Ref. 7, pp. 131–61)
22.and H. Rund [The Significance of Nonlinearity in the Natural Sciences, edited by B. Kursunoglu, A Perlmutter, and L. F. Scott (Plenum, New York, 1977), p. 121–43].
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