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# Classical Dynamics - MAT00048I

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• Department: Mathematics
• Credit value: 20 credits
• Credit level: I
• Academic year of delivery: 2023-24
• See module specification for other years: 2024-25

## Module summary

The module aims to introduce some of the 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.

## Related modules

• None

### Prohibited combinations

• None

Pre-requisite modules:

• Introduction to Applied Mathematics
• Foundations & Calculus
• Multivariable Calculus & Matrices

Post-requisite modules:

• Quantum & Continuum Dynamics

This module is the first part of the Applied Mathematics stream, and as such must be taken with the second part (Quantum & Continuum Dynamics)

## Module will run

Occurrence Teaching period
A Semester 1 2023-24

## Module aims

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

## Module learning outcomes

By the end of the module, students will be able to:

1. Manipulate vectors and use them to formulate problems in dynamics, including many-body problems, and how to work within different frames of reference.

2. Seek conserved quantities such as energy and angular momentum, understand their relevance and relationships with equations of motion.

3. Solve the two-body gravitational problem and investigate the properties of orbits.

4. Work with reformulations of mechanics in the Lagrangian and Hamiltonian forms to provide appropriate frameworks for other theories of nature besides classical dynamics.

5. Analyse 1D and 2D dynamical systems (including classification of equilibrium points, bifurcations in 1D systems and phase portraits in 2D systems).

## Module content

Newtonian Gravitation

• Revision: Vectors, scalar and vector products and triple products, time-derivatives; index notation and summation convention.

• Newton’s Law of Motion: Frames of reference, Galilean relativity, Newton's laws. Energy, momentum, angular momentum. Circular motion and angular velocity. Collections of particles.

• Newtonian Gravity: Newton's law of gravity. Two-body problem, central forces and resulting planar motion in polar coordinates. The geometry of orbits: ellipses and Kepler's laws; parabolas, hyperbolas and scattering. Energy, effective potential, stability of orbits.

Lagrangian and Hamiltonian Dynamics

• 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 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, phase portraits

## Indicative assessment

Closed/in-person Exam (Centrally scheduled) 100

### Special assessment rules

None

There will be five formative assignments with marked work returned in the seminars. At least one of them will contain a longer written part, done in LaTeX.

### Indicative reassessment

Closed/in-person Exam (Centrally scheduled) 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy

S Strogatz, Nonlinear dynamics and Chaos, CRC Press

M Lunn, A first course in Mechanics, (Oxford University Press (U1 LUN)

TWB Kibble and FH Berkshire, Classical Mechanics, Imperial College Press (U1 KIB)

R Fitzpatrick, Newtonian Dynamics, Lulu (U1.3 FIT)

R Douglas Gregory, Classical Mechanics Cambridge University Press (U1 GRE)

P Smith and RC Smith, Mechanics John Wiley and Sons (U1 SMI )

H Goldstein, Classical Mechanics, Addison-Wesley, (U1 GOL). [Later editions in conjunction with C Poole and J Safko]

LN Hand and JD Finch, Analytical Mechanics, Cambridge University Press (U1.017 HAN).

NMJ Woodhouse, Introduction to Analytical Dynamics, Oxford University Press, (U1.3WOO)

The information on this page is indicative of the module that is currently on offer. The University constantly explores ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.