1. ## Vibration Analysis

Vibration analysis will probably be concerned with linear differential operators acting on some space of functions. These functions will probably be defined on some interval [a,b], and they will have an inner product $\langle f,g\rangle$, defined by $\langle f,g\rangle = \int_a^bf(t)g(t)\,dt$. A linear differential operator L acting on these functions will have an adjoint L*, defined by $\langle L^*f,g\rangle = \langle f,Lg\rangle$ for all f,g. Unsurprisingly, L is called selfadjoint if L*=L.