9
Normal modes
Any student of the physical sciences will encounter the subject of oscillations on
many occasions and in a wide variety of circumstances, for example the voltage
and current oscillations in an electric circuit, the vibrations of a mechanical
structure and the internal motions of molecules. The matrices studied in the
previous chapter provide a particularly simple way to approach what may appear,
at first glance, to be difficult physical problems.
We will consider only systems for which a position-dependent potential exists,i.e., the potential energy of the system in any particular configuration depends
upon the coordinates of the configuration, which need not be be lengths, however;
the potential mustnotdepend upon the time derivatives (generalised velocities) of
these coordinates. So, for example, the potential−qv·Aused in the Lagrangian
description of a charged particle in an electromagnetic field is excluded. A
further restriction that we place is that the potential has a local minimum at
the equilibrium point; physically, this is a necessary and sufficient condition for
stable equilibrium. By suitably defining the origin of the potential, we may take
its value at the equilibrium point as zero.
We denote the coordinates chosen to describe a configuration of the systembyqi,i=1, 2 ,...,N.Theqineed not be distances; some could be angles, for
example. For convenience we can define theqiso that they are all zero at the
equilibrium point. The instantaneous velocities of various parts of the system will
depend upon the time derivatives of theqi, denoted byq ̇i. For small oscillations
the velocities will be linear in the ̇qiand consequently the total kinetic energyT
will be quadratic in them – and will include cross terms of the formq ̇i ̇qjwith
i=j. The general expression forTcan be written as the quadratic form
T=∑i∑jaijq ̇iq ̇j= ̇qTA ̇q, (9.1)where ̇qis the column vector ( ̇q 1 ̇q 2 ··· ̇qN)T and theN×NmatrixA
is real and may be chosen to be symmetric. Furthermore,A, like any matrix