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XIII.—Partial Differential Equations and the Calculus of Variations

Published online by Cambridge University Press:  15 September 2014

E. T. Copson
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Extract

A partial differential equation of physics may be defined as a linear second-order equation which is derivable from a Hamiltonian Principle by means of the methods of the Calculus of Variations. This principle states that the actual course of events in a physical problem is such that it gives to a certain integral a stationary value.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 46 , 1927 , pp. 126 - 135
Copyright
Copyright © Royal Society of Edinburgh 1927

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References

page 126 note * See Emmy Noether, Göttinger Nachrichten (1918), p. 238. Copson, E. T., Proc. Edin. Math. Soc. (1923–1924), 42, pp. 61–68CrossRefGoogle Scholar.

page 126 note † It is, of course, possible to adopt in this work the Summation Convention of the Tensor Calculus, that when a Greek letter occurs twice in a term, once as index and once as suffix, then the terms obtained by giving all possible values to this Greek letter are to be summed. For the sake of clearness and simplicity this has, however, not been done.

page 127 note * See Schouten, Der Ricci-Kalkül, p. 25.

page 128 note * See Goursat, , Cours d'Analyse (third edition) (1923), 3, p. 147Google Scholar, on adjoint equations.

page 129 note * Goursat, , Cours d'Analyse, (third edition), (1918), 2, p. 620Google Scholar, et seq.

page 129 note † Copson, E. T., Proc. Edin. Math. Soc., 43, (1924–1925), pp. 35–38CrossRefGoogle Scholar.

page 130 note * There is no loss in generality in this supposition, except in the case when all the first minors of the vanishing determinant P vanish. This case is of no very great interest, however.

page 131 note * That Pαβ has this form follows from Bellavitis' Theorem on vanishing determinants. See Mem. Istituto, Venezia, 7 (1857).

page 132 note * Cf. Courant u. Hilbert, Methoden der Mathematischen Physik, i, p. 205, where we find the statement: “Wir erkennen also, dass elliptische Differentialgleichungen aus einem definiten, hyperbolische aus einem indefiniten und parabolische aus einem semi-definiten Integranden durch Variation entspringen.”

This statement is true only as regards the elliptic and hyperbolic cases and follows from the relation aβγ=pβγ. The semidefinite integrand leads to an equation apparently in n variables, but actually in fewer than n.