am1053 intro to mathematical modelling · 2. rate equations: simple mathematical models of systems...

AM1053

Intro to Mathematical Modelling

Lecturer: Prof. Sebastian Wieczorek

Contact: WGB G49, sebastian.wieczorek@ucc.ie

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?