Abstract
In dealing with a function of a single variable, y = f (x), in the ordinary calculus, we often find it of use to determine the values of x for which the function y is a local maximum or a local minimum. By a local maximum at position x 1, we mean that f at position x in the neighborhood of x 1 is less than f (x 1) (see Fig. 2.1). Similarly for a local minimum of f to exist at position x 2 (see Fig. 2.1) we require that f (x) be larger than f (x 2) for all values of x in the neighborhood of x 2. The values of x in the neighborhood of x 1 or x 2 may be called the admissible values of x relative to which x 1 or x 2 is a maximum or minimum position.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Courant: “Differential and Integral Calculus,” Interscience Press.
- 2.
For a rigorous study of this subject refer to “Calculus of Variations” by Gelfand and Fomin. Prentice-Hall Inc., or to “An Introduction to the Calculus of Variations,” by Fox, Oxford University Press.
- 3.
See the references cited earlier.
- 4.
This problem has been solved by both Johann and Jacob Bernoulli, Sir Isaac Newton, and the French mathematician L’Hôpital.
- 5.
We shall consider Hamilton’s principle in detail in Chap. 7.
- 6.
In seeking an optimal solution in a problem we strive to attain, subject to certain given constraints, that solution, amongst other possible solutions, that satisfies or comes closest to satisfying a certain criterion or certain criteria. Such a solution is then said to be optimal relative to this criterion or criteria, and the process of arriving at this solution is called optimization.
- 7.
Thus y(x) will correspond to “a” of the early extremization discussion of f(x) while \(\tilde y\)(x) corresponds to the values of x in the neighborhood of “a” of that discussion.
- 8.
For a particular function Ė(x) continuous in the interval (x 1, x 2), if \(\int_{{x_1}}^{{x_2}} {\phi (x)\eta (x)dx = 0}\) for every continuously differentiable function η(x) for which η(x 1) = η(x 2) = 0, then ϕ ≡ 0 for x 1 ≤ x ≤ x 2.
- 9.
Note that for a two parameter family we have from Eq. (2.22) the result:
$$\delta ^{(1)} I = \left( {\frac{{\partial \tilde I}}{{\partial \varepsilon _1 }}} \right)_{\begin{array}{*{20}c} {\varepsilon _1 = 0} \\ {\varepsilon _2 = 0} \\\end{array}} \varepsilon _1 + \left( {\frac{{\partial \tilde I}}{{\partial \varepsilon _2 }}} \right)_{\begin{array}{*{20}c} {\varepsilon _1 = 0} \\ {\varepsilon _2 = 0} \\\end{array}} \varepsilon _2 $$(2.25) - 10.
The Lagrange multiplier is usually of physical significance.
- 11.
In dynamics of particles, if the constraining equations do not have derivatives the constraints are called holomonic.
- 12.
In problems of solid mechanics dealing with the total potential energy we will see that the kinematic boundary conditions involve displacement conditions of the boundary while natural boundary conditions involve force conditions at the boundary.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Dym, C.L., Shames, I.H. (2013). Introduction to the Calculus of Variations. In: Solid Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6034-3_2
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6034-3_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6033-6
Online ISBN: 978-1-4614-6034-3
eBook Packages: EngineeringEngineering (R0)