This chapter presents the Laplace transform, which is as fundamental to continuous-time systems as the z-transform is to discrete-time systems. Several properties and examples are presented. Similar to the z-transform, the Laplace transform can be regarded as a generalization of the appropriate Fourier transform. In continuous time, the Laplace transform is very useful in the study of systems represented by linear constant-coefficient differential equations (i.e., rational LTI systems). Frequency responses, resonances, and oscillations in electric circuits (and in mechanical systems) can be studied using the Laplace transform. The application in electrical circuit analysis is demonstrated with the help of an LCR circuit. The inverse Laplace transformation is also discussed, and it is shown that the inverse is unique only when the region of convergence (ROC) of the Laplace transform is specified. Depending on the ROC, the inverse of a given Laplace transform expression may be causal, noncausal, two-sided, bounded, or unbounded. This is very similar to the theory of inverse z-transformation. Because of these similarities, the discussion of the Laplace transform in this chapter is brief.

This chapter is intended to review concepts that the reader has some familiarity with and introduce high level descriptions of linear marine systems analysis. An initial discussion on the similarity between mechanical vibration equations of motion and marine dynamical systems is made. Mechanical vibrations are defined as vibrations in the absence of fluids. Examples of static and dynamic coupling between the various modes of motion or degrees of freedom are presented. The differences between frequency domain and time domain representations are given by introducing the concept of response amplitude operators (RAO’s). Complex arithmetic and linear, second order differential equations are briefly reviewed. Two examples of mechanical vibrations that are relevant to marine dynamics are developed and solved. The first example has to do with base excitation, similar to what a high speed planing craft may experience in long waves. The second example addresses one method for vibration isolation/suppression, that may, or may not, be useful in shock/impact mitigation schemes.

Regarding the key components of macroeconomics as continuous (rather than discrete) and considering the corresponding dynamics leads to differential equations. In this chapter, we focus on first-order differential equations and consider two methods (applicable to certain types of equation): separation and the use of integrating factors. We look at economic applications to continuous-time price adjustment and continuous cash flows.

This project explores the phenomenon of coupled harmonic oscillators, which have a broad relevance in fields such as mechanics, atomic and quantum physics, optics, electronics, and biology. The paper expands on a model of a harmonic oscillator, using numerical methods for solving differential equations to analyse systems of coupled oscillators. The study focuses on two illustrative cases: a one-dimensional system of two point masses suspended from springs, and a system of two simple pendulums moving in the same plane. The derived mathematical equations of motion provide a comprehensive framework for understanding the behaviour of such systems. Through computational experimentation, the project aims to elucidate the oscillatory behaviour of these systems depending on different parameters and their coupling, particularly focusing on energy transfer between the oscillators. The Runge–Kutta algorithm is employed for solving the initial value problem (IVP) for the ordinary differential equations (ODE) governing these systems. The project underscores the versatility of the harmonic oscillator model by showing that different physical systems can be described by the same mathematical model.

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.

Classical continuum mechanics focuses on the deformation field of moving continua. This deformation field is composed of the trajectories of all material elements, labeled by their initial positions. This initial-condition-based, material description is what we mean here by the Lagrangian description of a fluid motion. In contrast to typical solid-body deformations, however, fluid deformation may be orders of magnitude larger than the net displacement of the total fluid mass. The difficulty of tracking individual fluid elements has traditionally shifted the focus in fluid mechanics from individual trajectories to the instantaneous velocity field and quantities derived from it. These quantities constitutethe Eulerian description of fluids. This chapter surveys the fundamentals of both the Lagrangian and the Eulerian approaches. We also cover notions and results from differential equations and dynamical systems theory that are typically omitted from fluid mechanics textbooks, yet are heavily used in later chapters ofthis book.

During the seventeenth century, the advent of what were known as the “common” and “new” analyses fundamentally changed the landscape of European mathematics. The widely accepted narrative is that these analyses, analytic geometry and calculus (mostly due to Descartes and Leibniz, respectively), occasioned a transition from geometrical to symbolic methods. In dealing with the science of motion, mathematicians abandoned the language of proportion theory, as found in the works of Galileo, Huygens, and Newton, and began employing the Newtonian and Leibnizian calculi when differential and fluxional equations first appeared in the 1690s. This was the advent of a more abstract way of practicing mathematics, which culminated with the algebraic approach to calculus and mechanics promoted by Euler and Lagrange in the eighteenth century. In this chapter, it is shown that geometrical interpretations and mechanical constructions still played a crucial role in the methods of Descartes, Leibniz, and their immediate followers. This is revealed by the manner in which they handled equations and how they sought their solutions. The passage from proportions to equations did not occur in a single step; it was a process that took a century to reach completion.

The differential equations as written by Leibniz and by his immediate followers look very similar to the ones in use nowadays. They are familiar to our students of mathematics and physics. Yet, in order to make them fully compatible with the conventions adopted in our textbooks, we have to change just a few symbols. Such “domesticating” renderings, however, generate a remarkable shift in meaning, making those very equations – when so reformulated – not acceptable for their early-modern authors. They would have considered our equations, as we write them, wrong and corrected them back, for they explicitly adopted tasks and criteria different from ours. In this chapter, focusing on a differential equation formulated by Johann Bernoulli in 1710,I evaluate the advantages and risks inherent in these anachronistic renderings.

When most students first approachfeedback control, they are still coming to grips with its foundation inlinear system theory. This unsteadiness with thefoundationmakes understanding feedback dramatically moredifficult. Experienced professionals canhavesimilar problems, painfully compounded by the fog of imperfect memory of the linear system theorystudied years before. It is thusvery common for all types of students to sit down to study feedback, glimpse the breadth and depth of knowledge required as a prerequisite, and simply give up. The purpose of this chapter is to strengthen the reader in linear systemfundamentals. It will fill in the gaps for those who need gaps filled,deepen the understanding of thosefluent in the mechanics of solving problems but who were nevershown the overarching conceptual logic, and serve as a handy reference for those who are truly comfortable.

After a brief discussion of invariant measures and entropy, we introduce semidynamical and dynamical systems. We use Koopmanism to show how to obtain semigroups/groups of linear operators on function spaces, when we have a quasi-invariant measure. Applications are given to solution flows of differential equations. In the last part we discuss “dilations” as a mathematical approach to the origins of irreversibility.