Environmental Engineering Reference
In-Depth Information
forces us to recognise that our intuitive ideas about how things move are often
incorrect. At the most fundamental level, mechanics of either the classical or the
quantum kind, in either the relativistic or non-relativistic limit, is a study of motion
and to study motion is to ask some fundamental questions about the nature of space
and time. In this topic we will draw out explicitly the different underlying structures
of space and time used in the approaches of Newton and Einstein.
1.1.1 Space and the classical particle
We all have strong intuitive ideas about space, time and motion and it is precisely
because of this familiarity that we must take special care in our attempts to define
these fundamental concepts, so as not to carry too many unrecognised assumptions
along with us as we develop the physics. So let us start by picking apart what
we mean by position. We can usually agree what it means for London to be
further away than Inverness and we all know that in order to go to London from
Inverness we must also know the direction in which to travel. It may also seem
to be fairly uncontentious that an object, such as London, has a position that
can be specified, i.e. it is assumed that given enough information there will be no
ambiguity about where it is. Although this seems reasonable, there is immediately a
problem: day-to-day objects such as tennis balls and cities have finite size; there are
a number of 'positions' for a given object that describe different parts of the object.
Having directions to London may not be enough to find Kings Cross station, and
having directions to Kings Cross station may not be enough to find platform number
nine. To unambiguously give the position of an object is therefore only possible if
the object is very small - vanishingly small, in fact. This hypothetical, vanishingly
small object is called a particle. It might be suggested that with the discovery of the
substructure of the atom, true particles, with mass but no spatial extent, have been
identified. However, at this level, the situation becomes complicated by quantum
uncertainty which makes the simultaneous specification of position and momentum
impossible. The classical particle is therefore an idealisation, a limit in which the
size of an object tends to zero but in which we ignore quantum phenomena. Later
we shall see that it is possible to define a point called the centre of mass of an
extended object and that this point behaves much like a classical particle. The
collection of all possible positions for a particle forms what we call space.
The mathematical object possessing the properties we require for the description
of position is called the vector. A vector has both magnitude and direction and we
must be careful to distinguish it from a pure number which has a magnitude, but no
directional properties. The paradigm for the vector comes from the displacement of
a particle from point
A
to point
B
as shown in Figure 1.1. The displacement from
A
to
B
is represented by the directed-line-segment
AB
. We can imagine specifying
the displacement as, for example, “start at
A
and move 3 km to the northeast”
or “start at
A
and go 1 parsec in the direction of Alpha Centuri”. Once we have
specified a displacement between the two points
A
and
B
we can imagine sliding
each end of the line segment in space until it connects another two points
C
and
D
.
To do this, we move each end through the same distance and in the same direction,
