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On Professor Whittaker's solution of differential equations by definite integrals: Part I

Published online by Cambridge University Press:  20 January 2009

W. O. Kermack  and
W. H. McCrea
W. O. Kermack
Affiliation:
(Royal College of Physicians, Edinburgh)
W. H. McCrea
Affiliation:
(Edinburgh University)
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In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 2 , Issue 4 , June 1931 , pp. 205 - 219
Copyright
Copyright © Edinburgh Mathematical Society 1931

References

page 205 note 1 The theory is given explicitly for a single pair of variables q, p (or Q, P). It is clear, however, that it may be generalised at once to include any number of pairs.

page 205 note 2 Professor Whittaker takes both signs positive; but it is more convenient for the present method to take one positive and one negative.

page 209 note 1 CfWhittaker, and Watson, , Modern Analysis (1927), 353 (Ex. 11)Google Scholar

page 216 note 1 This is a particular case of a general result. CfWhittaker, and Watson, , Modern Analysis (1927), p. 385, Ex. 50. It should be noted thatGoogle Scholar

page 219 note 1 Zs. f. Physik, 40 (1927), 809.CrossRefGoogle Scholar