Coupled Oscillators without Damping—Lagrange’s Equations

Abstract

Having discussed electrical networks in rather general terms, we might appear to be needlessly repeating now when we begin a new discussion of coupled mechanical oscillators without damping. Indeed, the systems that now interest us are a subset of those treated previously, but we shall discuss them in a rather different way. Instead of asking for the response of a system to a given force F(t), we shall attend mainly to the undriven motion of a set of coupled undamped oscillators, and our principal interest will be the ways in which such a system can move in simple harmonic motion. These ways of moving, known as normal modes of motion, have their own characteristic normal frequencies; they are important in various application of quantum theory—in particular, the theory of molecular vibrations and the theory of vibrations of crystal lattices, and, by extension, the quantum theory of fields. Although we shall not be applying quantum theory, we can in the context of classical mechanics carry out the part of such calculations that is common to classical and quantum theory. The mathematical methods and the physical ideas encountered in this work are much the same in classical as in quantum theory.