A note on small vibrations

Abstract

This note exhibits some old results in what is possibly a new form.The simple harmonic oscillator, whose kinetic and potential energies are given in terms of a single coordinate x bywhere 6 is a positive constant, has the equation of motionThe solution, appropriate to the initial conditions x = x 0) x =x 0 at t = 0 is conveniently writtenThe problem of the small oscillations of a conservative dynamical system of n degrees of freedom about a position of stable equilibrium is a generalisation of the problem of the oscillator; in seeking a generalisation of the solution (1) a matrix notation at once suggests itself. Let the dynamical system be defined by the two quadratic formswhere B is a symmetric square matrix, x a column matrix and x' its transpose. The form T is of course positive definite, and F is assumed to be non-negative. The equations of motion areThe standard method of solution is to find the latent roots p\, pi p\ of the matrix B, which will for the moment be assumed positive and distinct, and to construct the matrix L whose columns are the normalised latent vectors. Then L'BL is a diagonal