This chapter exploresthe Fermi–Pasta–Ulam–Tsingou (FPUT) problem in the context of a one-dimensional chain of interacting point masses. We start by modelling the system as a chain of masses interacting through a force dependent on their relative displacements. Next, we simplify this system to harmonic oscillators under a linear force dependency, further developing it to describe a wave-like behaviour. The chapter discusses dispersion relations and the impact of boundary conditions leading to discretisation of allowed wave modes. A non-linear, second-order interaction is then included, complicating the system’s dynamics and necessitating the use of numerical methods for its solution. We then track the system’s evolution in a multidimensional phase space, leading to observations of seemingly chaotic motion with emerging periodicity. Energy conservation and its flow through the system are crucial aspects of the analysis. A detailed numerical procedure is provided, involving solution of initial value problems, mode projections, and energy computations to explore the complex behaviour inherent in the FPUT problem.

This chapter addresses the eigenvalue problem (EVP) with a focus on its application to describing standing waves or stationary quantum systems. A numerical method known as the shooting method is introduced to solve the EVP. Using a cylindrical waveguide (e.g., optical fibre) as a system model, the normal modes of a scalar wave are explored. The wave equation is examined, with considerations made for axial symmetry, boundary conditions, and the influence of refraction coefficients. A significant part of the study is devoted to finding the normal modes and associated wave numbers in an optical fibre. The latter part of the chapter presents the shooting method, a recursive technique to ascertain the eigenvalue in numerical calculations. The applicability of this method is further examined in the context of a quantum well. This chapter offers a thorough exploration of the EVP, highlighting its relevance to real-world research and introducing a robust numerical method for its resolution.

Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.

The results for the vibrating string are generalized to discuss vibrating systems that are not necessarily harmonic. Solutions are characterized by counting the nodes for solutions—that is, locations where the vibrational amplitude is always zero. One-dimensional solutions for uniform bars and rods, relevant for various chimes and bars used for music, are presented. The nonuniform bars used for xylophones and marimbas, and how they are used for tuning, are presented. The procedure is generalized for two-dimensional objects, where node lines are considered, and applied to membranes, such as drum heads, and plates. The complications that arise with degeneracies, or multiple solutions with the same frequency, are presented. A general, symbolic way of representing vibrational modes is shown, as well as how that can, very symbolically, represent any possible solution for a vibrating object as a sum of those with sinusoidal time dependence.

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.