Hostname: page-component-f554764f5-nqxm9 Total loading time: 0 Render date: 2025-04-09T09:59:57.910Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-timing, variational principles and waves

Published online by Cambridge University Press:  29 March 2006

G. B. Whitham
G. B. Whitham
Affiliation:
California Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 44 , Issue 2 , 11 November 1970 , pp. 373 - 395
Copyright
© 1970 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Waltham, Mass.: Blaisdell.
Kuzmak, G. E. 1959 Prikl. Mat. Mekh. Akad. Nauk S.S.S.R. 23, 515. (Translated in Appl. Math. Mech. 23, 73.)
Landau, L. D. & Lifshitz, E. M. 1960 Mechanics. Oxford: Pergamon.
Luke, J. C. 1966 Proc. Roy. Soc. A 292, 403.
Whitham, G. B. 1965a Proc. Roy. Soc. A 283, 238.
Whitham, G. B. 1965b J. Fluid Mech. 22, 27.
Whitham, G. B. 1967a J. Fluid Mech. 27, 39.
Whitham, G. B. 1967b Proc. Roy. Soc. A 299, 6.