This project focuses on the Initial Value Problem (IVP) for ordinary differential equations with the application of multipoint recursion schemes. The effectiveness and convergence of these schemes are explored and subsequently applied to examine the properties of a compound pendulum, specifically the dependence of the oscillation period on energy. The chapter then focuses on Newton’s laws of motion, laying the foundation for understanding the motion equation. The project uses a simple pendulum to illustrate the concept, looking at how changes in amplitude affect the period of harmonic oscillations. Numerical methods, such as recursive methods based on local extrapolation, are then employed to derive formulas. The project concludes by discussing the integration of Runge–Kutta methods and implicit schemes to solve the equations. This project ultimately questions the viability of the pendulum as a standard unit of time, adding value to ongoing discussions in physics and mathematics education.

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 presents all the needed theoretical background regarding the initial value problem for a first order ordinary differential equation in finite dimensions. Local and global existence, uniqueness, and continuous dependence on data are presented. The discussion then turns to stability of solutions. We discuss the flow map and the Alekseev-Grobner Lemma. Dissipative equations. and a discussion of Lyapunov stability of fixed points conclude the chapter.