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# MATH1057 Dynamics and Relativity

## Module Overview

This module is designed to introduce students to central elements of applied mathematics. It assumes no prior knowledge of particular applications, but assumes a working understanding of basic vector algebra and simple differential equations. The module provides the foundation more advanced applied mathematics as well as an interesting self-contained module for students who do not wish to take other applied mathematics modules subsequently. The module focuses on Newtonian and relativistic dynamics, two of the great intellectual developments in scientific history. These provide beautiful and accurate descriptions of physical situations from the human to astronomical scales, for bodies travelling at speed up to (near) that of light. The last part of the module introduces a systematic beautiful mathematical treatment that prepares the way for the user to study the dynamics of more complicated contexts such as general relativity. Applications include fluid flow, rockets, fairground rides, Halley’s comet, spacecraft, the weather and even doing your laundry! One of the prerequisites for MATH2044, MATH3006, MATH3072 and MATH6149

### Aims and Objectives

#### Module Aims

The module aims: • to give the student an introduction to basic concepts in Newtonian mechanics, without assuming any prior knowledge; • to give the student the basic mathematical skills to analyse problems involving the motion of particles in three dimensions.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• understand the concepts of relative velocity and of gravitational and frictional forces
• derive and solve the differential equations arising from Newton’s laws of motion applied to simple situations using vector notation
• understand the fundamental concepts of special relativity
• understand and exploit the conservation laws of momentum and energy for simple situations;
• derive key results for time dilation, length contraction and velocity addition at high speed
• recall and apply the concept of the Lagrangian in simple situations

### Syllabus

Vectors: Revision of vector algebra General motion: position/velocity/acceleration components Newtonian kinematics: Forces and energy Galilean transformation Time/space/events/observers Relative velocity Special relativity: Spacetime Lorentz transformation Velocity addition Time dilation/length contraction Redshift four-momentum and E=mc2 Newtonian dynamics: Centre of mass/equilibrium Newton’s laws Simple forces: gravity, friction, elasticity Conservation of mass, linear momentum Collisions (including relativistic effects) Rocket motion Rigid body motion Cylindrical and spherical polar coordinates Orbital motion Analytical dynamics: The calculus of variations Lagrangians The Euler-Lagrange equations

### Learning and Teaching

#### Teaching and learning methods

The lecturer will provide a structured week-by-week study programme, based largely on the notes provided. Each week there will be three hours of lectures. There will be a problems class every week, each lasting one hour which will be used to study problems illustrating the lecture material. Students may also attend a workshop each week to gain additional personal help with understanding lecture material or problem sheets. Students should spend their private study time studying the lecture notes and working through these problem sets. Online learning materials are also provided via the a Blackboard website.

TypeHours
Independent Study96
Lecture36
Tutorial12
Practical classes and workshops6
Total study time150

#### Resources & Reading list

Gregory D.R.. Classical Mechanics.

### Assessment

#### Summative

MethodPercentage contribution
Class Test 10%
Weekly Homeworks 20%
Written exam  (2 hours) 70%

#### Referral

MethodPercentage contribution
Written exam 100%

#### Repeat Information

Repeat type: Internal & External

### Linked modules

Pre-requisites: MATH1048 AND MATH1059

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