PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations
Conditions of Validity, Ch Sturm-Liouville theory: the spectral theorem for a differential operator
Problem
A uniform heavy chain of length L hangs vertically, fixed at the top (x=L) and free at the bottom (x=0), where x is measured upward from the free end. Its small transverse normal modes Y(x) satisfy dxd(xdxdY)+λY=0,0<x≤L, where λ=ω2/g, ω is the mode's angular frequency and g is the gravitational acceleration. Near the free end, the equation has two independent solution behaviours: a bounded one, Y1(x)→c (a finite constant, with Y1′(x) tending to a finite limit) as x→0, and an unbounded one, Y2(x)∼Alnx (with Y2′(x)∼A/x) as x→0, for constants c=0, A=0.
((a)) Write the chain's equation in Sturm-Liouville form and identify p(x), q(x) and w(x). Show that x=0 is a singular endpoint while x=L is regular.
((b)) State the boundary condition to impose at the regular end x=L (fixed top) and explain why no condition of the form αY(0)+βY′(0)=0 is imposed at x=0.
((c)) Evaluate p(x)Y1′(x) and p(x)Y2′(x) as x→0, and hence show that admitting Y2 into the domain makes the boundary form p(Y1Y2′−Y1′Y2) fail to vanish at x=0, while retaining only bounded multiples of Y1 makes it vanish for every pair in the domain. State which solution describes a physical hanging chain and why.