The Applied Module in Stage 2 aims to introduce some of the main ideas and theories of modern applied mathematics and mathematical physics, along with some of the main mathematical methods that are used to study and solve problems in these theories. Rather than present the methods in isolation, the aim is to encounter them in the context of applications, so that theory and technique progress in tandem. The overall aim is to lay the foundations for the further study of applied mathematics and mathematical physics in Stages 3 and 4.
As part of these broad aims, this module has the following components:
Introduction to Dynamical Systems and Newtonian Gravity (Autumn) provides an introduction to key methodological techniques for the analysis of dynamical systems, illustrated by examples, working up from low dimensions to implications in higher dimensions. It then moves on the the development of Newton’s theory of motion in vectorial form, leading up to the description of orbits in Newtonian gravity. Dynamical systems is further developed in the Waves and Fluids components as well as in various course in years 3 and 4, while Newtonian mechanics is further developed in the Classical Dynamics, Quantum Dynamics and Waves and Fluids components.
ClassicalDynamics (Spring) presents a sophisticated form of Newton’s laws known as analytical mechanics, which also forms an important component of modern theories of both classical and quantum physics.
Quantum Dynamics (Spring/Summer) begins the development of quantum mechanics, and various relevant techniques of differential equations which have numerous other applications in pure and applied mathematics.
Waves and Fluids (Spring/Summer), exploring the the dynamics of continuous media, focusing on elementary fluid dynamics and the motion of waves. This lays the foundations for the full development of fluid dynamics in stages 3 and 4, as well as for modules on electromagnetism and quantum mechanics. The mathematical techniques of vector calculus are employed and further developed, as are Fourier methods.
Studying these four components alongside each other during the course of the year will allow students to see the many connections across different areas of Applied Mathematics; understanding these connections and being able to use ideas and techniques across many contexts is an essential part of the modern mathematician’s toolkit.
Module learning outcomes
Introduction to Dynamical Systems:
Flows on a line: 1D Equations and exact solutions, dimensionless form, fixed points, stability
Bifurcations in 1D: Fold, trans-critical and pitchfork bifurcations
Flows in 2D: linear systems, classification, linearisation of non-linear systems, introduction to phase portraits
·Revision: Vectors, scalar and vector products and triple products, time-derivatives
·Frames of reference, Galilean relativity, Newton's laws. Energy, momentum, angular momentum. Circular motion and angular velocity.
·Many particles, two particles, Newton's law of gravity. Central forces and resulting planar motion in polar coordinates. The geometry of orbits: ellipses and Kepler's laws; parabolae, hyperbolae and scattering. Energy, effective potential, stability of orbits.
Lagrangian mechanics. Constraints, generalized coordinates, Lagrange's equations, connection to Newton’s laws. Constants of the motion: ignorable coordinates and Jacobi's function. Qualitative analysis of systems with a single degree of freedom using Jacobi's function. Examples, including conservative central forces and the Lagrangian analysis of the spinning top.
Hamiltonian mechanics. Generalized momenta and the Hamiltonian. Derivation of Hamilton's equations from Lagrange's equations. Conservation results. Poisson brackets. Equations of motion and conservation laws in Poisson bracket form.
Phase-plane techniques. Trajectories and equilibria for conservative and damped systems.
Variational principles. Reformulation of Lagrangian mechanics (and Hamiltonian, if time permits) in variational form. Examples of other variational problems (e.g., brachistochrone, geodesics on a plane and sphere).
Introduction to quantum mechanics. Schroedinger's equation (motivated by brief discussion of Planck-Einstein and de Broglie relations). Time-independent Schroedinger equation. The one-dimensional box. Probability interpretation of the wavefunction and the orthogonality of distinct energy eigenfunctions.
Eigenvalue problems of Sturm-Liouville type Reality of eigenvalues, orthogonality of eigenfunctions for distinct eigenvalues, eigenfunction expansions. Applications, including the heat equation and the quantum mechanics of square wells and boxes in one and three dimensions.
Series solution methods (e.g., motivated by the one-dimensional harmonic oscillator) Power series solutions of first and second order equations. Legendre's equation, Hermite's equation etc. Regular singular points: the method of Frobenius. Application to the quantum harmonic oscillator.
Spherical harmonics and the hydrogen atom
Introduction to Waves
The 1D wave equation, particularly vibrations of a string. Waves on an infinitely long string: characteristics, D’Alembert’s formula. Concepts: standing waves, the wave number, the wave energy, wave packets. Waves on a semi-infinite string: boundary conditions and reflection of waves. Vibrations of a finite string: Fourier methods.
The 2D wave equation. Vibrations of rectangular and circular membranes. Vector calculus: operators in curvilinear coordinates, particularly polar coordinates. (Limited statements on Bessel functions.)
Shallow water waves: the hydraulic jump.
Elementary fluid dynamics
Continuum fields: density, velocity. Particle paths, streamlines and streaklines. Eulerian and Lagrangian descriptions of a continuum medium. Conservation of mass, incompressibility. 2D motion and the streamfunction.
The Euler momentum equation, surface and volume forces, pressure. Bernoulli’s theorem and applications.
Kinematic and dynamic conditions at a free surface. Surface gravity waves. (Limited statements on potential flow.)
Academic and graduate skills
Mathematics graduates are problem solvers with an ability to work from first principles and to employ diverse and appropriate techniques. This module helps develop these essential skills; students will learn material and techniques with a wide range of applications in modern descriptions of physical phenomena.
The understanding of motion provided by Newton has provided key insights into the physical universe, allowed technological progress through engineering, and has driven developments in mathematics including calculus and differential geometry.
Quantum mechanics has revolutionized the modern world through electronic, optical and nuclear technology, and new quantum technologies are emerging in communication and computation. This module provides a first introduction to this theory, but importantly, also develops the study of differential equations that has applications in many other branches of pure and applied mathematics. Analytical mechanics provides a basis for many modern theories of physics and has also led to developments in geometry as well as the field of chaotic dynamics.
Fluid and wave phenomena are ubiquitous in nature and industry with important applications such as meteorology, ocean dynamics, biofuels, aeronautics, astrophysics, diseases of the cardiovascular system and the swimming of plankton and whales. Wave phenomena appear in many other systems, from musical instruments to tsunamis.