14 Small Oscillations I

Ashok Goyal

1. Introduction

So far we considered the motion of a particle with few degrees of freedom in a central – force field and solved the Lagrange’s equation of motion to study its dynamical evolution in space and time. If we have a complex system in which many particles are coupled together with forces it is clear that the coordinate of any one particle will depend on the behavior of the coordinates of other particles and the problem in general would be quite complicated to visualize. It will be however, possible to make a transformation from Cartesian coordinates with simple time dependence. In the case of the system executing small oscillations, these generalized coordinates would oscillate with single well defined frequency. Such coordinates are called ‘Normal Coordinates’ and for suitable initial conditions on Cartesian coordinates and velocities, the subsequent motion will indeed take place with a single frequency. The general solution of the system would indeed entail a combination of the Normal Mode Solutions.

2. Lagrangian formulation of the system

We consider a dynamical system with  degrees of freedom specified by the generalized coordinates . It is assumed that the system is conservative such that the potential energy of the system U does not depend on time explicitly and is a function of generalized coordinates alone i.e.

This system is in equilibrium when the generalized forces acting on the system are zero i.e.

Potential energy thus has an extremism value in the equilibrium position specified by the set of generalized coordinates . The equilibrium will be stable if this extremum is a ‘local minima’ which means that a small disturbance from equilibrium will result in an increase in the potential energy and since the system is conservative, the total energy which is the sum of kinetic and potential energy remains constant, the kinetic energy would result in a decrease of velocity which will finally come to zero. The system would thus have bound motion about its equilibrium position. In the case of unstable equilibrium, the potential energy function has a local maxima and any deviation from equilibrium would result in increase in velocity and will give rise to unstable motion.

In what follows, we will be interested in the dynamics of the system near its stable equilibrium position. In other words, we will confine ourselves to the discussion of the system for small deviations from its equilibrium position. This has a very important characteristic. Even if we do not have any detailed knowledge of the potential energy function, in the limit of small departures, the Taylor series expansion of the potential function U allows as to consider a small departure from equilibrium to consider the force arising from the potential to be harmonic. Thus for small departures, the system will always execute motion as if the interaction is simple harmonic.

Let the deviation of the generalized coordinate  from the equilibrium is denoted by :

where we have made the Taylor expansion of the potential function about the equilibrium position. From the equilibrium condition (14.2), we can write

The Lagrange’s equation of the motion is given by

And we have a set of n second – order linear homogeneous differential equations with constant coefficients.

3. Eigen values and Eigen functions

Equation (14.4) has an oscillatory solution of the form

Where we have used the symmetric nature of

This equation is called the ‘characteristic equation’ or ‘secular equation’ of the system. It is an equation of degree n in  and has, in general, n roots. The ’s are called ‘eigen frequencies’ of the system. If more than two frequencies’ are equal, they are called ‘degenerate’.

Since there are n frequencies , for each frequency we have a set ’s. Each set can be considered to define the components of a n-dimension vector  called the eigen-vector of the system. Thus  is the eigen-vector associated with the frequency . If we write  and A as matrices with components  and  respectively, the equation (14.16) can be written in the matrix form

By the principle of superposition, the general solution for  would be a superposition of all solutions for each of the n values of the frequency

4. The Orthogonality of the Eigen-vectors

The eigenvectors  form an orthogonal set can be seen as follows:

Eqn. (14.16) for the sth root corresponding to  of the characteristic equation is given by

The equation for the rth root is

Using the symmetry property of  and , we can write this as

Having normalized the sum and combining with (14.25) we have the orthonormality condition of the eigenvectors namely,

5. Normal coordinates

The general solution eqn. (14.20) for the motion of the coordinate  is the sum over terms each of which depends on the individual eigen frequency. The coefficients are normalized according to eqn.(14.27). We can rewrite the solutions by multiplying with a constant scale factor  which in general is complex. Thus

And there are  such independent coordinates and the equations of motion expressed in these normal coordinates are completely separable. Note that

Thus, the system expressed in terms of normal coordinates render the potential and kinetic energies of the system in the diagonal form and the motion of the system completely separates into independent motions of the normal coordinates each oscillating with its eigen-frequency. In order to completely specify the transformation to normal coordinates, we will require the specification of complex quantities ’s introduced in the definitions of normal coordinates. These can be completely determined from the initial conditions on the coordinates ’s and velocities ’s at.

6. Summary

• If a conservative dynamical system in the presence of a potential field makes a small departure from its equilibrium position, the system would execute motion as if the interaction is Simple Harmonic. This is independent of the exact nature of the potential field.
• A simple transformation from Cartesian coordinates to what are known as Normal Coordinates allows the description of the system in these coordinates which oscillate with single well defined frequencies.