Modern Classical Mechanics

In the preceding chapter on rigid-body motion we took a step beyond single-particle mechanics to explore the behavior of a more complex system containing many particles bonded rigidly together. Now we will explore additional sets of many-particle systems in which the individual particles are connected by linear, Hooke’s-law springs. These have some interest in themselves, but more generally they serve as a model for a large number of coupled systems that oscillate harmonically when disturbed from their natural state of equilibrium, such as elastic solids, electric circuits, and multi-atom molecules. We will begin with the oscillations of a few coupled masses and end with the behavior of a continuum of masses described by a linear mass density. The mathematical techniques required to analyze such coupled oscillators are used throughout physics, including linear algebra and matrices, normal modes, eigenvalues and eigenvectors, and Fourier series and Fourier transforms.