This section may be omitted.) The previous section is of fun-
Damental importance, for it is an introduction to the methods of
Mathematical physics, as they are applied to dynamic systems.
The reader is therefore strongly advised to work through all the
Exercises, for only in this way can a real grasp of the principles be
Obtained. If he has done this, he will be better equipped to appre-
Ciate the meaning of this section, which summarises the method.
The physicist starts by naming his variables— x 1, x2, … xn. The
Basic equations of the transformation can then always be obtained
By the following fundamental method:—
Take the first variable, x1, and consider what state it will
Change to next. If it changes by finite steps the next state will be
x1' if continuously the next state will be x1+ dx1. (In the latter case
He may, equivalently, consider the value of dx1/dt.)
Use what is known about the system, and the laws of phys-
ics, to express the value of x1', or dx1/dt (i.e. what x1 will be) in
terms of the values that x1, …, xn (and any other necessary factors)
Have now. In this way some equation such as
x1' = 2 αx 1 – x3 or dx1/dt = 4k sin x3
Is obtained.
35
34
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
TH E D ET ERM IN A TE MA C HI N E
Repeat the process for each variable in turn until the whole
Transformation is written down.
The set of equations so obtained— giving, for each variable in
The system, what it will be as a function of the present values of
The variables and of any other necessary factors— is the canonical
Representation of the system. It is a standard form to which all
Descriptions of a determinate dynamic system may be brought.
If the functions in the canonical representation are all linear, the
System is said to be linear.
Given an initial state, the trajectory or line of behaviour may
Now be computed by finding the powers of the transformation, as
In S.3/9.
*Ex. 1: Convert the transformation (now in canonical form)
dx/dt = y
dy/dt = z
dz/dt = z + 2xy– x 2
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To a differential equation of the third order in one variable, x. (Hint: Elimi-
Nate y and z and their derivatives.)
*Ex. 2: The equation of the simple harmonic oscillator is often written
D 2x
-------- + ax = 0
Dt 2
Convert this to canonical form in two independent variables. (Hint: Invert
The process used in Ex. 1.)
*Ex. 3: Convert the equation
D 2x2 dx2
-x -------- – ( 1 – x ) ----- + -------------- = 0
2dt 1 + x 2dt
To canonical form in two variables.
Unsolvable” equations. The exercises to S.3/6 will have
Shown beyond question that if a closed and single-valued transfor-
Mation is given, and also an initial state, then the trajectory from
That state is both determined (i.e. single-valued) and can be found
By computation For if the initial state is x and the transformation
T, then the successive values (the trajectory) of x is the series
X, T(x), T2(x), T3(x), T4(x), and so on.
This process, of deducing a trajectory when given a transforma-
Tion and an initial state, is, mathematically, called “integrating”
The transformation (The word is used especially when the trans-
Formation is a set of differential equations, as in S.3/7; the process
Is then also called “solving” the equations.)
If the reader has worked all through S.3/6, he is probably
Already satisfied that, given a transformation and an initial state,
He can always obtain the trajectory. He will not therefore be dis-
Couraged if he hears certain differential equations referred to as
“nonintegrable” or “unsolvable”. These words have a purely tech-
Nical meaning, and mean only that the trajectory cannot be
Obtained i f one is restricted to certain defined mathematical oper-
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Ations. Tustin’s Mechanism of Economic Systems shows clearly
How the economist may want to study systems and equations that
Are of the type called “unsolvable”; and he shows how the econo-
Mist can, in practice get what he wants.
Phase space. When the components of a vector are numerical
Variables, the transformation can be shown in geometric form, and
This form sometimes shows certain properties far more clearly and
Obviously than the algebraic forms that have been considered so far.
As example of the method, consider the transformation
x' = 1/2x + 1/2y
y' = 1/2x + 1/2y
After this discussion of differential equations, the reader who
Is used to them may feel that he has now arrived at the “proper”
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