Classical Mechanics

KEY FEATURES

The key features of this chapter are the existence of small oscillations near a position of stable equilibrium and the matrix theory of normal modes. A simpler account of the basic principles is given in Chapter 5.

Any mechanical system can perform oscillations in the neighbourhood of a position of stable equilibrium. These oscillations are an extremely important feature of the system whether they are intended to occur (as in a pendulum clock), or whether they are undesirable (as in a suspension bridge!). Analogous oscillations occur in continuum mechanics and in quantum mechanics. Here we present the theory of such oscillations for conservative systems under the assumption that the amplitude of the oscillations is small enough so that the linear approximation is adequate. A simpler account of the theory is given in Chapter 5. This treatment is restricted to systems with two degrees of freedom and does not make use of Lagrange's equations. Although the material in the present chapter is self-contained, it is helpful to have solved a few simple normal mode problems before.

The best way to develop the theory of small oscillations is to use Lagrange's equations. We will show that it is possible to approximate the expressions for T and V from the start so that the linearized equations of motion are obtained immediately. The theory is presented in an elegant matrix form which enables us to make use of concepts from linear algebra, such as eigenvalues and eigenvectors.