Differential equations and dynamical systems are essential mathematical tools for modeling a large variety of phenomena. They are used in Physics, Chemistry, Biology, Ecology, Finance, Sociology, and beyond, to describe various processes. However, differential equations in most cases cannot be solved and we need to use qualitative methods in order to understand such processes. In this course we will do two things. First, we will see how to solve some first-order differential equations and linear differential equations. The fact that such differential equations can be solved makes them very special. Second, we will consider nonlinear differential equations. In general, these cannot be solved, and therefore we will need to qualitatively understand the behavior of the solutions, without actually finding them. To achieve this, we will combine our knowledge of the solutions of linear differential equations with other theoretical results from the theory of dynamical systems. This course uses results from single- and multi-variable Calculus and from Linear Algebra and it is a prime example of how such results can be applied in more complicated contexts with a wide range of applications in other scientific domains.
The course starts with refreshing and improving basic mathematical skills. Then real functions of one real variable are treated: standard and composed functions, models, limits, derivatives, extrema, Taylor expansions, integration (applications and techniques). Subsequently complex numbers are introduced. Finally, differential equations are solved to describe processes in physics, economy, medical sciences, etc. This topic includes: initial value problems, directional fields, equilibrium and stability, separation of variables, variation of constants, homogeneous and inhomogeneous systems, Euler’s method, conversion of complex solutions into real-valued solutions. The material is illustrated through examples that focus on the applications of mathematical techniques.
Complex analysis treats complex valued functions defined in the complex plane which are analytic, i.e., differentiable. These include functions from calculus, such as exponential, sine, cosine, logarithm and square root, but also polynomials, quotients of polynomials, and functions which can be composed out of these. Analytic functions have nice intrinsic properties, which sometimes help to better understand properties of real-valued functions. An important aspect of the course is the treatment of the calculus of residues, by means of which integrals can be evaluated.
The course gives an overview of topics on Complexity and Networks. In the first part, consisting of four lectures, we discuss synchronization and the stability of synchronized solutions for systems connected in a network.
Problems in physics are in general modeled in terms of PDEs such as the wave equation, the diffusion equation, and the Laplace equation. In this course the basic notions for PDEs will be treated, including classification of PDEs, uniqueness of solutions, and well-posedness. The course will focus on analytical methods to solve PDEs, including the method of characteristics, separation of variables, Green’s functions, and Fourier series.
Foundations of classical mechanics: Newton’s laws of motion, inertial and non-inertial reference frames, energy principles. Applications to simple dynamical systems under various force systems. Newton’s law of gravitation and its application to motions of planetary bodies and the orbits of satellites. Motion relative to a rotating frame, coriolis and centripetal forces, motion under gravity over the earth’s surface. Rigid body dynamics: centre of mass, angular velocity and momentum principles. Plane motions of laminae, simple 3-dimensional rigid body motions with reference to practical examples such as the orbiting space station, and the axis of rotation of the earth. Introduction to Lorentz transformations (time permitting).
Wave particle duality. Schrödinger’s equation for simple one dimensional systems. Finite-dimensional Hilbert space and matrix mechanics. Angular momentum and the hydrogen atom. Perturbation theory and the variational method. Collapse of the wave function, Schrödinger’s cat and the EPR paradox.
In Euclidean spaces, continuity is defined using the familiar Euclidean distance between points. In this course we generalize the concept of continuity first to metric and then to topological spaces. In metric spaces we are given the concept of distance between its “points”. Function spaces are some of the most useful examples of metric spaces. In topological spaces we know only which subsets are open. It turns out that this is all we need in order to study continuity. We discuss further some important analytical and geometrical concepts such as compactness, connectedness, and completeness.
Gauss and Riemann, among others, studied the properties of manifolds: curved, non-Euclidean spaces. But it was Einstein’s theory of General Relativity that familiarized physicists with such spaces. Differential forms are a powerful tool for computing on manifolds, while at the same time they offer a beautiful geometric abstraction. After introducing the concept of a submanifold of a Euclidean space we discuss vector fields and differential forms in Euclidean space. Finally, we generalize these concepts to manifolds. We show how the generalized Stokes theorem expressed in the language of differential forms unifies the, well-known from vector calculus, theorems of Gauss and Stokes. Finally, we discuss applications of differential forms in electromagnetism and dynamical systems.