Literature

• I.M. Gelfand, S.V. Fomin, “Calculus of Variations” (Sec. 1-5, 7).
• J.D. Logan, “Applied Mathematics” (Sec. 3.1 - 3.5).
• Fowles & Cassiday, "Analytical Mechanics" (Chapter 10).
• H. Goldstein, “Classical Mechanics” (Chapter 2).

Functionals [GF Sec 1.1?]

A functional assigns a number to each function in a class. Thus, it is a function where the independent variable is itself a function.

Examples

1. Associate its length to each rectifiable curve.
2. The intergal of a non-negative function $y(x)$ defined in an interval $[x_1, x_2]$

$$ J[y] = \int_{x_1}^{x_2} y(x)\, dx $$

gives the area between the $x$ axis and the graph of the function.

1. For a continuously differentiable function $y(x)$ define the number

$$ J[y] = \int_a^b y'^2(x)\,dx. $$