Applied Mathematics at Oxford

spreadeaglerainMechanics

Oct 24, 2013 (3 years and 11 months ago)

156 views

Applied Mathematics at Oxford

Christian Yates

Centre for Mathematical Biology

Mathematical Institute

Who
am
I?



Completed my
B.A. (Mathematics) and M.Sc. (Mathematical
M
odelling

and
S
cientific
C
omputing) at
the Mathematical Institute as a member of
Somerville
College.


Currently completing my D.Phil. (
M
athematical Biology) in the Centre for
Mathematical Biology as a member of Worcester and St. Catherine’s colleges.


Next year


Junior
R
esearch
F
ellow at Christ Church college.


Research in
cell migration, bacterial motion and locust motion.


Supervising
Masters students.


Lecturer at Somerville College


Teaching
1st and 2nd year tutorials in
college.

Outline
of this talk


The principles of applied mathematics


A practical example


Mods

applied mathematics (first year)


Celestial
mechanics


Waves on strings



Applied mathematics options (second and third year)


Fluid mechanics


Classical mechanics


Mathematical Biology


Reasons to study mathematics

Outline
of this talk


The principles of applied mathematics


A simple example


Mods applied mathematics (first year)


Celestial mechanics


Waves on strings



Applied mathematics options (second and third year)


Fluid mechanics


Classical mechanics


Calculus of variations


Mathematical Biology


Reasons to study mathematics

Principles
of applied mathematics



Start from a physical or “real world” system


Use physical principles to describe it using mathematics


For example, Newton’s Laws


Derive the appropriate mathematical terminology


For example, calculus


Use empirical laws to turn it into a solvable mathematical
problem


For example, Law of Mass Action, Hooke’s Law


Solve the mathematical model


Develop mathematical techniques to do this


For example, solutions of differential equations


Use the mathematical results to make predictions about the real world system

Simple
harmonic motion


Newton’s second law


Force = mass x acceleration






Hooke’s Law


Tension = spring const. x extension




Resulting differential equation

simple harmonic motion


Re
-
write in terms of the displacement from equilibrium





which is the description of simple harmonic motion


The solution is




with constants determined by the initial displacement and velocity


The period of oscillations is



Putting maths to the test: Prediction


At equilibrium (using Hooke’s law T=
ke
):





Therefore:




So the period should be:




Experiment

Equipment:


Stopwatch


Mass


Spring


Clampstand


1 willing volunteer



Not bad but not perfect



Why not?


Air resistance


Errors in measurement etc


Old Spring


Hooke’s law isn’t perfect etc


Outline
of this talk


The principles of applied mathematics


A simple example


Mods

applied mathematics (first year)


Celestial mechanics


Waves on strings



Applied mathematics options (second and third year)


Fluid mechanics


Classical mechanics


Mathematical Biology


Reasons to study mathematics

Celestial
mechanics


Newton’s 2nd Law





Newton’s Law of Gravitation





The position vector satisfies the differential
equation




Solution of this equation confirms Kepler’s Laws

How long is a year?


M=2x10
30
Kg


G=6.67x10
-
10
m
3
kg
-
1
s
-
2


R=1.5x10
11
m






Not bad for a 400 year old piece of
maths
.




Kepler

Outline
of this talk


The principles of applied mathematics


A simple example


Mods

applied mathematics (first year)


Celestial mechanics


Waves on strings



Applied mathematics options (second and third year)


Fluid mechanics


Classical mechanics


Mathematical Biology


Reasons to study mathematics

Waves
on a string


Apply Newton’s Law’s to each
small interval of string...



The
vertical
displacement satisfies
the partial differential equation






Known as the wave
equation


Wave speed:


Understanding music


Why don’t all waves sound

like this?




Because we can superpose waves on each other

=


By adding waves of different amplitudes and frequencies we can come up with
any shape we want:





The

maths

behind how to find the correct signs and amplitudes is called
F
ourier

series analysis.

Fourier series

More complicated wave forms


Saw
-
tooth wave:





Square wave:

Outline
of this talk


The principles of applied mathematics


A simple example


Mods

applied mathematics (first year)


Celestial mechanics


Waves of strings



Applied mathematics options (second and third year)


Fluid mechanics


Classical mechanics


Mathematical Biology


Reasons to study mathematics

Fluid
mechanics


Theory of flight
-

what causes the lift on an aerofoil?








What happens as you cross the sound barrier?

Outline
of this talk


The principles of applied mathematics


A simple example


Mods

applied mathematics (first year)


Celestial mechanics


Waves of strings



Applied mathematics options (second and third year)


Fluid mechanics


Classical mechanics


Mathematical Biology


Reasons to study mathematics

Classical
mechanics


Can we predict the motion of a double
pendulum?








In principle

yes.


In practice, chaos takes over.







Outline
of this talk


The principles of applied
mathematics


A simple example


Mods

applied mathematics (first
year)


Celestial mechanics


Waves of strings



Applied mathematics options
(second and third year)


Fluid mechanics


Classical mechanics


Mathematical Biology


Reasons to study mathematics

H
ow
we do mathematical
biology?


Find out as much as we can
about the biology


Think about which bits of our
knowledge are important


Try to describe things
mathematically


Use our mathematical
knowledge to predict what we
think will happen in the
biological system


Put our understanding to good
use

Mathematical
biology

Locusts

Switching behaviour


Locusts switch direction periodically


The

length of time between switches depends on the density of the group

30 Locusts

60 Locusts

Explanation
-

Cannibalism

Outline
of this talk


The principles of applied mathematics


A simple example


Mods

applied mathematics (first year)


Celestial mechanics


Waves on strings



Applied mathematics options (second and third year)


Fluid mechanics


Classical mechanics


Calculus of variations


Mathematical Biology


Reasons to study mathematics

Why
mathematics?


Flexibility
-

opens many
doors


Importance
-

underpins science


Ability to address fundamental questions about the universe


Relevance to the “real world” combined with the beauty of abstract theory


Excitement
-

finding out how things work


Huge variety of possible
careers


Opportunity to pass on knowledge to others

Me on Bang goes the theory

I’m off to watch Man City in the FA cup final

Further
information


Studying mathematics and joint schools at
Oxford


http://www.maths.ox.ac.uk


David Acheson’s page on dynamics


http://home.jesus.ox.ac.uk/~dacheson/
mechanics.html


Centre for Mathematical Biology


http://www.maths.ox.ac.uk/groups/math
ematical
-
biology/


My web page


http://people.maths.ox.ac.uk/yatesc/