Introduction to Mathematical Physics/Energy in continuous media/Exercises

Exercice: Find the equation evolution for a rope clamped between two walls.

Exercice: {{{1}}}

Exercice: Give the expression of the deformation energy of a smectic (see section secristliquides for the description of smectic) whose i$^{t}h$ layer's state is described by surface $u_{i}(x,y)$,

Exercice: Consider a linear, homogeneous, isotrope material. Electric susceptibility $\epsilon$ introduced at section secchampdslamat allows for such materials to provide $D$ from $E$ by simple convolution:

eqexsusc

$D=\epsilon *E.$

where $*$ represents a temporal convolution. To obey to the causality principle distribution $\epsilon$ has to have a positive support. Indeed, $D$ can not depend on the future values of $E$. Knowing that the Fourier transform of function "sign of $t$" is $C.V_{p}(1/x)$ where $C$ is a normalization constant and $V_{p}(1/X)$ is the principal value of $1/x$ distribution, give the relations between the real part and imaginary part of the Fourier transform of $\epsilon$. These relations are known in optics as Krammers--Kr\"onig relations\index{Krammers--Kr\"onig relations}.