Mathematical Methods for Physics and Engineering - Matematica.NET

9Normal modesAny student of the physical sciences will encounter the subject of oscillations onmany occasions and in a wide variety of circumstances, for example the voltageand current oscillations in an electric circuit, the vibrations of a mechanicalstructure and the internal motions of molecules. The matrices studied in theprevious 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 dependsupon the coordinates of the configuration, which need not be be lengths, however;the potential must not depend upon the time derivatives (generalised velocities) ofthese coordinates. So, for example, the potential −qv · A used in the Lagrangiandescription of a charged particle in an electromagnetic field is excluded. Afurther restriction that we place is that the potential has a local minimum atthe equilibrium point; physically, this is a necessary and sufficient condition forstable equilibrium. By suitably defining the origin of the potential, we may takeits value at the equilibrium point as zero.We denote the coordinates chosen to describe a configuration of the systemby q i , i =1, 2,...,N.Theq i need not be distances; some could be angles, forexample. For convenience we can define the q i so that they are all zero at theequilibrium point. The instantaneous velocities of various parts of the system willdepend upon the time derivatives of the q i , denoted by ˙q i . For small oscillationsthe velocities will be linear in the ˙q i and consequently the total kinetic energy Twill be quadratic in them – and will include cross terms of the form ˙q i˙q j withi ≠ j. The general expression for T can be written as the quadratic formT = ∑ ∑a ij ˙q i ˙q j = ˙q T A˙q, (9.1)i jwhere ˙q is the column vector (˙q 1 ˙q 2 ··· ˙q N ) T and the N × N matrix Ais real and may be chosen to be symmetric. Furthermore, A, like any matrix316