User:Bob K31416/CoV

Calculus of variations

Calculus of variations is concerned with variations of functionals. A functional depends on a function, i.e. has a function as its argument. The value of a functional is a real number. A variation of a functional is a small change in the functional's value due to a small change in the function that is its argument. The first variation is defined as the linear part of the change in the functional and the second variation is defined as the quadratic part. For example, if $J[y]$ is a functional with the function $y = y(x)$ as its argument, and there is a small change in its argument from $y$ to $y + h$, where $h = h(x)$ is a function in the same function space as $y$, then the corresponding change in the functional is


 * $$ \Delta J[h] = J[y+h] - J[y] $$.

The functional $J[y]$ is said to be differentiable if
 * $$ \Delta J[h] = \phi [h] + \epsilon \|h\| $$ ,

where $&phi;[h]$ is a linear functional, $&phi;[h]$ is the norm of $&phi;[&alpha;h] = &alpha; &phi;[h]$, and $&phi;[h_{1} +h_{2}] = &phi;[h_{1}] + &phi;[h_{2}]$ as $h, h_{1}, h_{2}$. The linear functional $&alpha;$ is the first variation of $||h||$ and is denoted by,


 * $$\delta J[h] = \phi(h) $$.

The functional $h$ is said to be twice differentiable if


 * $$ \Delta J[h] = \phi_1 [h] + \phi_2 [h] + \epsilon \|h\|^2 $$ ,

where $h(x)$ is a linear functional (the first variation), $a &le; x &le; b$ is a quadratic functional, and $h(x)$ as $||h||$. The quadratic functional $|h(x)|$ is the second variation of $a &le; x &le; b$ and is denoted by,


 * $$\delta^2 J[h] = \phi_2(h) $$.

The second variation $&epsilon; &rarr; 0$ is said to be strongly positive if
 * $$\delta^2J[h] \ge k \|h\|^2 $$ ,

for all $||h|| &rarr; 0$ and for some constant $&phi;[h]$.

Sufficient condition for a minimum:

The functional $J[y]$ has a minimum at $J[y]$ if its first variation $&phi;_{1}[h]$ at $&phi;_{2}[h]$ and its second variation $&epsilon; &rarr; 0$ is strongly positive at $||h|| &rarr; 0$.

Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero.

The interest is in extremal functions that make the functional attain a maximum or minimum value, or more generally stationary functions, which are those functions where the variation (or differential) of the functional is zero.

Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. More generally, it is concerned with stationary values of functionals and the functions that make them stationary, which by definition is when the variation (or differential) of the functional is zero. Functionals are often expressed as definite integrals involving functions and their derivatives.

Normed linear spaces,
 * $$ \mathcal {C}(a,b) $$, consisting of all continuous functions defined on a closed interval [a,b]
 * $$ \mathcal {D}_1(a,b) $$, consisting of all continuous functions defined on a closed interval [a,b] that have continuous first derivatives.


 * $$ \mathcal {D}_1(a,b) \subset \mathcal {C}(a,b) $$

Norm of function y in space $$ \mathcal {C} $$
 * $$ \| y \|_0 = \max_{a \le x \le b} |y(x)| $$

Norm of function y in space $$ \mathcal {D}_1 $$
 * $$ \| y \|_1 = \max_{a \le x \le b} |y(x)| + \max_{a \le x \le b} |y'(x)| $$

An admissible function or curve is one that satisfies the constraints of a given variational problem.

An extremal function is a function which makes a given functional stationary.

An extremal function makes the variation of a functional vanish. Extremal functions are sometimes called stationary functions. The value of a functional when its variation vanishes is called a stationary value.


 * field of mathematical analysis
 * branch of the mathematical field of functional analysis
 * finds stationary values of functionals
 * finds functions that make the functional stationary
 * concerned with arbitrarily small changes in functionals, which are called variations or differentials
 * maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers.
 * extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero.
 * set of functions for a given variational problem are those satisfying certain constraints and are called admissible functions
 * an admissible function that makes a functional have a maximum or minimum value is called an extremal.
 * maxima and minima of functionals are called extrema
 * Functionals are often expressed as definite integrals involving functions and their derivatives.
 * arbitrarily small changes in functionals or functions are called variations or differentials





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