Example: the Pythagoras theorem
As a first example we shall prove the Pythagoras
Theorem using only arguments from dimensional
analysis.
Let ABC be a rectangular triangle with hypotenuse c
(facing C) and other two sides a (facing A) and b (facing
B), and let CH be the perpendicular from C to c. The
interior angle at A is denoted by (see picture).
Remark: Note and remember that angles are
dimensionless quantities (because they are defined via
the ratio
length of an arc of circle/length of radius of circle)!!
This means also that any functions of angles are
dimensionless quantities: for example, the trigonometric
functions and the inverse trigonometric functions.
Let S be the area of ABC. Obviously, S is uniquely
determined by c and :
S S (c, ).
The angle is dimensionless, and so we can also write
S c 2 ( ), (1)
2
because c has the dimension of area, just like S, i.e., the
2
ratio S/ c is dimensionless. Moreover, if the triangle is
non-degenerate then S>0, therefore, also ( ) 0
holds.
Now consider ACH and CBH. They are both
rectangular, too, and are similar to ABC (see the
picture). The sum of the areas of these two triangles is
equal to S, in other words, using (1) for each of the three
triangles gives us
c 2 ( ) a 2 ( ) b 2 ( )
and after dividing both sides by ( ) 0 we get the
proof of the Pythagoras theorem!
Examples from population dynamics
We will formulate here some simple models for
population dynamics in a closed ecosystem. Such models
play an important role when it comes to understanding
the factors that contributes to increase in population. Let
P=P(t) denote the size of the population at time t and P0
denote the size of the population at time t=0.
The simplest model is known as Malthus' model and it
says that the increase in population is proportional to the
size of the population, that is
where r is a constant speed of growth. This model
predicts that the population will increase exponentially,
that is
This would lead to a population explosion. Is this model
reasonable? As the population grows, the amount of
food, living space and natural resources will limit the
growth. We should therefore correct Malthus' model with
a limiting term. The growth depends not only on the size
of the population but also on how far it is from its upper
limit. We must correct with a term that makes the growth
decrease and becomes zero when we reach this upper
limit K. We have thus reached the logistic model (also
known as Verhulst's population model)
where K is a constant that can be interpreted as the
largest amount of individuals that an ecosystem can
nourish (the so called carrying capacity).
Now we scale this model (see the part of today’s lecture
on scaling). We introduce a dimensionless time
and a dimensionless population
We can now write the logistic equation in these
dimensionless variables
that is
where
is dimensionless. This differential equation can for
instance be solved by separation of variables
We easily realize that
that is
The constant solution P=K is called attractor and has the
property that, independently of the size of the initial
population, the size of the population will tend to K as
the time goes to infinity.