Calculation of normal frequencies of a normal mode

Question

In the Classical mechanics book by Goldstein, it is stated that if one wants to find the normal frequencies of a system, $\omega$ then the following equation has to be solved:

$\left|\hat{V}-\omega^2\hat{T} \right| = 0$, where $\hat V$ and $\hat T$ are the potential energy and kinetic energy matrices respectively.

Everywhere in literature, I've seen the normal modes have been calculated in the aforesaid way (like in Wikipedia, Classical Mechanics book by K. Symon). But now I'm getting confused while going through the lecture notes by Prof. Emanuele Berti (Page $28$, below the equation $3.3.44$).

What is being done here is that, he found the Wronskian $(W)$ of the differential equation governing the system (here a finite string), and then found the roots of $W=0$. He then claimed that the roots are the normal frequencies! (Quasi-normal modes are special type of normal modes with complex frequency. Here also the quasi-normal frequencies are calculated solving the $W=0$.)

How is this claim related to our formal knowledge about normal frequency: $\left|\hat{V}-\omega^2\hat{T} \right| = 0$ $?$

My observation: Wronskian is zero at the normal frequencies, means the solutions of the differential equations at the normal frequencies are linearly dependent. So how this linear dependence comes from $\left|\hat{V}-\omega^2\hat{T} \right| = 0$ $?$

Moreover $\left|\hat{V}-\omega^2\hat{T} \right| = 0$ is a consequence of the fact that the normal modes of a system satisfy the equation of motion of a classical simple harmonic oscillator with frequency $\omega$. But the solutions of the EOM of a classical SHO are linearly independent!

What am I missing?

Answer 1

The method described in Berti's notes is being reversely understood. What he has done is the conventional way of finding normal modes.

1. Give the Fourier Transform (w.r.t. time) to the differential equation that governs the system.
2. Solve the transformed equation along with the boundary conditions in order to find the normal frequencies, $\omega$.

The case, that is depicted in the notes, finds the $\omega$'s in the similar way. Now while giving the Inverse Fourier Transform to get the actual perturbation, one have to perform a contour integration (which has been said in the notes in case of an vibrating string), and in order to do that integration, he/she has to find the poles of the integrand. The poles of the solution(s) of a inhomogeneous linear differential equation often lies at the singularities of the corresponding Green's function (which in this case indicates the zeros of the Wronskian). So when the author tries to find the poles of the integrand function, we discovers that the Wronskian is getting vanished at the normal frequencies; this is just a coincidence.

Solutions of $W=0$ shouldn't necessarily imply the normal frequencies. But the point that is to noted is that, a vibrating system is a continuous mass system (this loosely means infinite number of infinitesimal mass elements and hence infinite number of normal modes). So the notion of Fourier Transform is coming.

But the equation $\left|\hat{V}-\omega^2\hat{T} \right| = 0$ is valid in case of discrete mass system (such as a simple pendulum or coupled pendulum or a triatomic molecule; where the number of particles becomes the total number of normal modes of the system). So here a harmonic time dependent solution $(\eta_j = Ca_j e^{-i\omega t})$ is chosen to reduce the mother differential equation: $T_{ij} \ddot{\eta_j} + V_{ij} \eta_j = 0$ (Goldstein's notation is being used). This is the basic difference between the discrete and continuous mass system.

Quasi-normal modes are also being calculated following the same prescription.