Four identical pendulums

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Invariance, Ch Representations, Schur's lemma and characters

Problem

Four identical pendulums, each of length

1.00\ \mathrm m

and mass

m

, hang so that at rest the bobs sit at the corners of a square, labelled

1,2,3,4

counterclockwise. Each swings at right angles to the plane of the square, with sideways displacement

x_i

. Bobs along an edge are joined by identical taut springs, and the two diagonal pairs by a second kind of taut spring. With the coupling idealised as for the three pendulums of this chapter,

\ddot{\mathbf x}=-K\mathbf x

with

K=\begin{pmatrix}a&b&c&b\\ b&a&b&c\\ c&b&a&b\\ b&c&b&a\end{pmatrix},\qquad a=\omega_0^2+\frac{2\kappa_1+\kappa_2}{m},\quad b=-\frac{\kappa_1}{m},\quad c=-\frac{\kappa_2}{m},

where

\omega_0^2=9.81\ \mathrm{s^{-2}}

\kappa_1/m=2.0\ \mathrm{s^{-2}}

for the edge springs and

\kappa_2/m=0.5\ \mathrm{s^{-2}}

for the diagonal ones. The symmetry operations of the square permute the four labels and form the group

C_{4v}

, with the axis through the centre of the square at right angles to it; the

\sigma_v

planes pass through opposite bobs and the

\sigma_d

planes through the midpoints of opposite edges. The character table on

(E,2C_4,C_2,2\sigma_v,2\sigma_d)

A_1=(1,1,1,1,1)

A_2=(1,1,1,-1,-1)

B_1=(1,-1,1,1,-1)

B_2=(1,-1,1,-1,1)

E=(2,0,-2,0,0)

. ((a)) Find the character of the representation of

C_{4v}

(x_1,x_2,x_3,x_4)

, and decompose it. ((b)) Find the normal modes and their values of

\omega^2

. Say which equality among these values is forced by the symmetry for every choice of

\kappa_1

and

\kappa_2

, and why. ((c)) Find the ratio

\kappa_2/\kappa_1

at which three normal modes share one frequency. Is that triple degeneracy forced by

C_{4v}

or accidental? Justify your answer with the degeneracy theorem.

Browse the problem library