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 layer's state is described by surface ,

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


where represents a temporal convolution. To obey to the causality principle distribution has to have a positive support. Indeed, can not depend on the future values of . Knowing that the Fourier transform of function "sign of " is where is a normalization constant and is the principal value of distribution, give the relations between the real part and imaginary part of the Fourier transform of . These relations are known in optics as Krammers--Kr\"onig relations\index{Krammers--Kr\"onig relations}.