User:BRousselet/sandbox

Why generalized eigenvalues?
In the entry, applications of eigenvalues and eigenvectors we find numerous scientific fields in which eigenvalues are used to obtain solutions. Generalized_eigenvalue_problem are less widespread but are a key in the study of Vibrations. They are useful when we use the Galerkin method  or  Rayleigh-Ritz method to find approximate solutions of partial differential equations modeling vibrations of structures such as strings and plates; the paper of Courant (1943) is fundamental. The Finite element method is a widespread particular case.

In classical mechanics, we may find generalzed eigenvalues when we look for vibrations of multiple degrees of freedom systems close to equilibrium; the kinetic energy provides the mass matrix $$ M $$, the potential strain energy provides the rigidity matrix $$ K $$. To get details, for example see the first section of this article (1941, in French)

With both methods, we obtain a system of differential equations or Matrix differential equation $$ M \ddot x+B \dot x +Kx=0 $$ with the mass matrix $$ M $$, the damping matrix $$ B $$ and the rigidity matrix $$ K $$. If we neglect the damping effect, we use $$ B=0$$, we can look for a solution of the following form $$ x=e^{i \omega t} u$$; we obtain that $$u $$ and $$\omega^2 $$ are solution of the generalized eigenvalue problem $$ -\omega^2 M u+Ku =0 $$

liens utiles
https://en.wikipedia.org/wiki/Finite_element_method_in_structural_mechanics https://en.wikipedia.org/wiki/Method_of_mean_weighted_residuals https://en.wikipedia.org/wiki/Galerkin_method#Matrix_form_of_Galerkin's_equation https://encyclopediaofmath.org/wiki/Ritz_method https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Galerkine https://fr.wikipedia.org/wiki/Valeur_propre,_vecteur_propre_et_espace_propre#Th%C3%A9orie_spectrale https://fr.wikipedia.org/wiki/Mode_normal https://en.wikipedia.org/wiki/Normal_mode#Standing_waves https://de.wikipedia.org/wiki/Eigenmode#Normalkoordinaten https://fr.wikipedia.org/wiki/Probl%C3%A8me_aux_valeurs_propres_g%C3%A9n%C3%A9ralis%C3%A9 https://en.wikipedia.org/wiki/Eigenfunction#Schr%C3%B6dinger_equation https://en.wikipedia.org/wiki/Rayleigh_quotient#Special_case_of_covariance_matrices https://en.wikipedia.org/wiki/Min-max_theorem https://en.wikipedia.org/wiki/Rayleigh%27s_quotient_in_vibrations_analysis https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem https://en.wikipedia.org/wiki/Semi-differentiability#Higher-dimensional_case