Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
AM1053
Intro to Mathematical Modelling
Lecturer: Prof. Sebastian Wieczorek
Contact: WGB G49, [email protected]
Lectures: Mon 10am WGB 107, Tue 10am WGB G03
All module information:
https://euclid.ucc.ie/Sebastian/teaching/AM1053.html
What is Applied Mathematics?
Mathematics has been used to describe real-world phenomena
already for thousands of years. However, the real breakthrough
took place in the 17th Century with Newtonian Dynamics.
Sir Isaac Newton (1642-1727) asked questions about how the
universe works. At that time, such questions were very
philosophical, in that they were studied by philosophers. What
Newton did was very unique and innovative: he created a whole
new language, called calculus, to study the universe in a
rational and precise way.
In 2005, Newton was deemed by the Royal Society to be more
influential than Albert Einstein! His theory is about 330 years
old (1687), but still remains an important subject that
influences modern science.
Following Newton’s theory, mathematics became an important
tool in many other areas of science, well beyond physics, and
changed the way we understand the world:
Biology (brain, neural networks, infectious diseases)
Information Technology (computers, lasers, internet)
Climate Science and Ecology (evolution, species adapting
to changing climate, chaos and unpredictability)
Machine Learning (Data + Computer Science + Dynamics,
e.g. self-driving cars)
Module Outline
1. Newtonian Dynamics: builds on AM1052 and focuses on
dynamics for mechanical systems: e.g. oscillating masses on
strings and springs.
2. Rate Equations: simple mathematical models of systems
evolving over time; e.g. population growth in ecology.
3. Modern Dynamics with Applications: introduces ideas
from Dynamical Systems Theory (e.g. phase space) and their
applications. It leads to AM2052 and AM3065.
1. Newtonian Dynamics
1.1 Basic Ideas
Modelling: While the world is a complicated place, applied
mathematics favours simple models. To make progress as
applied mathematicians, we often need to make choices and
simplify things. Simple models are not bad as long as they
include the relevant e↵ects (e.g. the key physical processes).
1. Gravity
2. Reaction forces
3. Friction
4. Air resistance
5. Flexing of rails/support
6. Rotation of Earth
7. Relativistic e↵ects
8. Quantum-mechanical e↵ects
Example: A Roller-coaster
1–4 A↵ects the motion. 6–8 Negligible E↵ects.
Approach: We start with the simplest picture that captures the
essential behaviour (1–3). Then, we incorporate more
complicated phenomena (4–5) as needed to improve the model.
Q: Is the project feasible? 1–3 should be enough to decide.
Q: How to design a safe roller-coaster? Need many more details,
this is usually the job for an engineer.
Point particles and centre of mass: Sticking to the simplest
models we approximate all bodies as point particles, that is
particles having zero size but non-zero mass.
This approximation works for planets, projectiles,
roller-coasters, bungee jumpers, etc. But why?
Archimedes showed that, given an extended body, there is a
special point called the centre of mass which moves as though
all the mass of the extended body were concentrated at that
point.
Note: the point particle description fails in many cases, for
example for falling leaves.
The Workflow
Physical System
Understand the inner workings and processes.
#
Write down mathematical equations to describe how the system
evolves over time.
#
Solve the mathematical problem.
#
Present and interpret the mathematical results.
What do they mean for the physical system at hand?
1.2 Kinematics: The description of motion
Coordinate system: We choose Euclidean geometry with three
perpendicular axes based at the origin O.
i is a vector of length one,
pointing in the direction of x;
j is a vector of length one,
pointing in the direction of y;
k is a vector of length one,
pointing in the direction of z.
Position
If the particle moves in time,
its time-dependent position
traces out a trajectory and is
described by a 3-dimensional
vector function of time:
r(t) = x(t)i+ y(t)j + z(t)k
Velocity and Speed
The velocity v(t) of a moving
body is the rate of change of its
position:
v(t) =d
dtr(t) = r(t)
Where one dot indicates one
time derivative.
v(t) = r(t) = x(t)i+y(t)j+z(t)k
The length of v(t), denoted
|v(t)|, gives the speed:
|v(t)| =p
x2 + y2 + z2
The direction of v(t) gives the
direction of motion; v(t) always
points along the trajectory.
Acceleration
The acceleration a(t) of a moving body is the rate of change of
its velocity v(t)
a(t) =d
dtv(t) =
d2
dt2r(t) = r(t),
where the double dot denotes the double time derivative.
a(t) = x(t)i+ y(t)j + z(t)k
Description of Motion
By “desribe the motion” we mean
• choose the coordinate system (if applicable),
• fix the initial time t0
(if applicable),
• obtain the position r(t), velocity v(t) and acceleration a(t).
Units are important to compare quantities:
r, x, y and z are measured in meters: m.
v and |v| are measured in meters per second:m
s.
a is measured in meters per second squared:m
s2.
m is measured in kilograms: kg.
Question: What is larger, one meter or one second?