Introduction to Mathematical Physics/Energy in continuous media/Exercises

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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^th 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 know in optics as Krammers--Kr\"onig relations\index{Krammers--Kr\"onig relations}.