Calculation of the variations

In functional Analysis, the calculation of the variations (or variational calculation) is a whole of methods making it possible to determine the critical points or the extreme ones of Fonctionnelle S .

The application of the theories of Welsh, Abel and the Transformée of Laplace made it possible to make of it a whole profitable branch of mathematics. It finds many applications in Mathematical physics, like the variational principles or seeks it minimal surfaces, of curved brachistochrones and Géodésique S.

Variations first and second

August 1st

Equation of Jacobi

August 1st

Combined points and condition of Legendre

August 1st

Condition of Weierstrass

Let us return to the expression of the Intégrale

$W\; =\; \backslash \; int\_\; \{x\_0\}\; ^\; \{x\_1\}\; F\; (X,\; there,\; y\text{'})\; \backslash ;\; \backslash mathrm\; dx$

and let us consider a field $F$ the extreme ones being composed of a family of these curves to a parameter $\backslash \; alpha$. Each one of them naturally satisfies the equation of Euler-Lagrange:

$\backslash \; frac\; \{\backslash \; mathrm\; D\}\; \{\backslash \; mathrm\; dx\}\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; y\text{'}\}\; -\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; =\; 0$.

By adopting the parametric representation: $x\_0$, $x\_1$ and $\backslash \; alpha$, functions of $t$, $x\_0$ and $x\_1$ follow curves $C$ and $D$ when $t$ varies, and the variation of $W$ from one extreme to another is

$\backslash \; delta\; W\; =\; \backslash \; left\; \backslash \; left\; (F\; +\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; y\text{'}\}\; (Y\text{'}\; -\; y\text{'})\; \backslash \; right)\; \backslash \; delta\; X\; \backslash \; right\_\; \{0\; \backslash \; to\; 1\}$,

where $y\text{'}$ is the angular coefficient of the tangent to extreme and $Y\text{'}$ that of the tangent to the curve $C$ or $D$.

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