5 The Wave Equation and the Third Axiom
43
it is still a remarkable result. We will explicitly show this at the end of this chapter.
We can therefore state
Newtonian point mechanics, the number of particles involved in motion remains unknown.
The x a can now themselves become the centres of inertia of many small particles,
etc. Without further knowledge of the systems physical constitution, we can make no
comment about the number of physical particles in our Newtonian axiomatic system,
up to an infinitely large number of particles, or a continuous mass distribution:
Only an experiment can decide if we are dealing with a continuum or if we are dealing with
a certain number of particles involved in a motion.
In regard to a later application, we will once again write down the equations of
motion for a system of particles whose coordinates q a have here not been closely
defined,
d
dt
m a
d
dt
q a
=
b =a
F ba + F a ,
F ab = −F ba .
⎫
⎪ ⎬
⎪ ⎭
(57)
In the light of this, we now understand the connection between our N discrete oscillating single masses and the oscillations of an elastic rod. The positions of the equally
spaced N masses according to (30), x i =
L
N
i , can be seen as the coordinates of the
centres of inertia of an improved equally spaced distribution of mass. Springs work
between the masses. If we halve the single masses and keep the equal distribution,
we have to halve the distance. This way the working single springs are halved. We
explained earlier that the halved spring has the doubled force constant.
The quantity s i is in Eq. (31) the displacement at the position x i . We can therefore
write s i (t) = s(x i , t) and according to (31) following is then valid,
m
∂
2
∂t 2 s(x i , t) = D N [s(x i−1 , t) − 2s(x i , t) + s(x i+1 , t)] .
Here, we insert x i =
L
N
i and after a simple rearrangement we write for the distances
between the particles x = L/N , hence
m
∂ 2
∂t 2 s
L
N
i, t
= D N
s
L
N
i + x, t
− s
L
N
i, t
−
s
L
N
i, t
− s
L
N
i − x, t
.
According to our explanations above, we can in the light of the Newtonian postulates
make N as large as we wish. The value
L
N
i approaches every point on the length
of the rod, in other words we can instead take a continuous variable x with optional
precision. We therefore accordingly substitute
L
N
i −→ x and receive
m
∂
2
∂t 2 s(x, t) = D N
s(x + x, t) − s(x, t)
−
s(x, t) − s(x − x, t)
.
43
it is still a remarkable result. We will explicitly show this at the end of this chapter.
We can therefore state
Newtonian point mechanics, the number of particles involved in motion remains unknown.
The x a can now themselves become the centres of inertia of many small particles,
etc. Without further knowledge of the systems physical constitution, we can make no
comment about the number of physical particles in our Newtonian axiomatic system,
up to an infinitely large number of particles, or a continuous mass distribution:
Only an experiment can decide if we are dealing with a continuum or if we are dealing with
a certain number of particles involved in a motion.
In regard to a later application, we will once again write down the equations of
motion for a system of particles whose coordinates q a have here not been closely
defined,
d
dt
m a
d
dt
q a
=
b =a
F ba + F a ,
F ab = −F ba .
⎫
⎪ ⎬
⎪ ⎭
(57)
In the light of this, we now understand the connection between our N discrete oscillating single masses and the oscillations of an elastic rod. The positions of the equally
spaced N masses according to (30), x i =
L
N
i , can be seen as the coordinates of the
centres of inertia of an improved equally spaced distribution of mass. Springs work
between the masses. If we halve the single masses and keep the equal distribution,
we have to halve the distance. This way the working single springs are halved. We
explained earlier that the halved spring has the doubled force constant.
The quantity s i is in Eq. (31) the displacement at the position x i . We can therefore
write s i (t) = s(x i , t) and according to (31) following is then valid,
m
∂
2
∂t 2 s(x i , t) = D N [s(x i−1 , t) − 2s(x i , t) + s(x i+1 , t)] .
Here, we insert x i =
L
N
i and after a simple rearrangement we write for the distances
between the particles x = L/N , hence
m
∂ 2
∂t 2 s
L
N
i, t
= D N
s
L
N
i + x, t
− s
L
N
i, t
−
s
L
N
i, t
− s
L
N
i − x, t
.
According to our explanations above, we can in the light of the Newtonian postulates
make N as large as we wish. The value
L
N
i approaches every point on the length
of the rod, in other words we can instead take a continuous variable x with optional
precision. We therefore accordingly substitute
L
N
i −→ x and receive
m
∂
2
∂t 2 s(x, t) = D N
s(x + x, t) − s(x, t)
−
s(x, t) − s(x − x, t)
.