Chapter One
1.3. A Mighty Mathematical Tool : The Equation of Motion
In this section we focus our attention on the motion of the electron. The best way to find out
how a body moves is to establish an equation. Exactly that's what we are going to do!
As we
have seen in Sec.1.1 (Lorentz-Force), we can describe velocity and acceleration
as vectors. To lead you to the equation of motion let us start with a small warm
up.
The x, y and z components of the average velocity are defined
by:
From these components we get the vector of the
average velocity:
Our next step is to construct the
momentary velocity 
. What can we do? We have to
make the difference 
infinitesimally small. According to
Leibniz this is expressed as:
Thus we realize that
the momentary velocity coordinates can be described as the first derivations with respect to time
of the coordinates x(t), y(t) and z(t). Newton wrote this derivations in the
following way:
(1c.01)
A
dot above a character indicates the first derivation with respect to time.
Similar to this, we can understand the acceleration.
The average acceleration is defined by:
We proceed as
we did with the average velocity; assuming that the differences are tending against zero we get
the momentary velocity components vx(t), vy(t)
and vz(t). In analogy to the momentary velocity, we can write:
(1c.02)
We introduce the
momentary velocity components as the first time-derivations of the coordinates x(t), y(t) and z(t).
Thus we have
After this step we are able to interpret
Newton's famous Lex Secunda:
(1c.03)
Force 
and acceleration 
are
both vectors. The momentum 
is also a vector, given by
(1c.04)
Now you see why we suggested the warm up; our equation
of motion reads:
After introducing a set of mutually perpendicular
base vectors 
x , y , z of unit
length (orthonormal base vectors), we can write the vectors 
and 
as columns:
(1c.05)
At this point we are ready to finish our
excursion. But this doesn't mean you may forget all! Surely the best is trying to
memorize as much as possible. Later you will see why.
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