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12 - CALCULUS OF VARIATIONS

Published online by Cambridge University Press:  05 February 2012

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Summary

In a previous chapter it was shown how to find stationary values of functions of a single variable f(x), of several variables f(x, y,…) and of constrained variables f(x, y,…) subject to gi(x, y,…) = 0, (i = 1, 2, …, m). In all these cases the forms of the functions f and gi were known and the problem was one of finding suitable values of the variables x, y,…

We now turn to a different kind of problem, one in which there are not free variables which must be chosen in order to bring about a particular condition for a given function, but in which the functions are free and must be chosen to bring about a particular condition for a given expression which depends upon these functions.

To give a more concrete example of the type of question to be answered, we may ask the following. ‘Why does a uniform rope suspended between two points take up the shape it does? Why doesn't it hang in an arc of a circle or in the form of three sides of a rectangle? Is it possible to predict the shape in which it will hang, that is to find a function, y = y(x), that gives the vertical height of the rope as a function of horizontal position?’

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Mathematical Methods for the Physical Sciences
An Informal Treatment for Students of Physics and Engineering
, pp. 332 - 348
Publisher: Cambridge University Press
Print publication year: 1974

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  • CALCULUS OF VARIATIONS
  • K. F. Riley
  • Book: Mathematical Methods for the Physical Sciences
  • Online publication: 05 February 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139167550.013
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  • CALCULUS OF VARIATIONS
  • K. F. Riley
  • Book: Mathematical Methods for the Physical Sciences
  • Online publication: 05 February 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139167550.013
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • CALCULUS OF VARIATIONS
  • K. F. Riley
  • Book: Mathematical Methods for the Physical Sciences
  • Online publication: 05 February 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139167550.013
Available formats
×