Chapter 5
The Wave Equation and the Third Axiom
The Newtonian equations have a remarkable property. They are strictly a point
mechanics, but do not have atomistics as a condition. This is simply because
Newtonian equations do not observe the physical positions of particles, but the equations observe the geometrical centres of inertia. This we can clarify with an example.
Two particles with the masses m 1 and m 2 move along a straight path, e.g. along
the x-axis towards each other with the velocity v 1 and v 2 until they ‘collide’. We
assume the conservation of the kinetic energy, hence we observe an elastic collision
and denote the resulting velocities of the masses after the collision as v
1 and v
2 , so
that the conservation laws for momentum and energy are then
m 1 v 1 + m 2 v 2 = m 1 v
1 + m 2 v
2 ,
1
2
m 1 v
2
1 +
1
2
m 2 v
2
2 =
1
2
m 1 v
1
2 +
1
2
m 2 v
2
2 .
(48)
Equation (48) after a simple calculation can be also be formulated as
m 1 (v
1 − v 1 ) = m 2 (v
2 − v 2 ) ,
m 1 (v
1 + v 1 ) (v
1 − v 1 ) = m 2 (v
2 + v 2 ) (v
2 − v 2 ) .
(49)
At first, Eq. (49) leads us to
v 1 + v
1 = v 2 + v
2
(50)
and we then find after a bit of calculation the result that can be found in every physics
textbook. The velocities after a collision are
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_5
39
The Wave Equation and the Third Axiom
The Newtonian equations have a remarkable property. They are strictly a point
mechanics, but do not have atomistics as a condition. This is simply because
Newtonian equations do not observe the physical positions of particles, but the equations observe the geometrical centres of inertia. This we can clarify with an example.
Two particles with the masses m 1 and m 2 move along a straight path, e.g. along
the x-axis towards each other with the velocity v 1 and v 2 until they ‘collide’. We
assume the conservation of the kinetic energy, hence we observe an elastic collision
and denote the resulting velocities of the masses after the collision as v
1 and v
2 , so
that the conservation laws for momentum and energy are then
m 1 v 1 + m 2 v 2 = m 1 v
1 + m 2 v
2 ,
1
2
m 1 v
2
1 +
1
2
m 2 v
2
2 =
1
2
m 1 v
1
2 +
1
2
m 2 v
2
2 .
(48)
Equation (48) after a simple calculation can be also be formulated as
m 1 (v
1 − v 1 ) = m 2 (v
2 − v 2 ) ,
m 1 (v
1 + v 1 ) (v
1 − v 1 ) = m 2 (v
2 + v 2 ) (v
2 − v 2 ) .
(49)
At first, Eq. (49) leads us to
v 1 + v
1 = v 2 + v
2
(50)
and we then find after a bit of calculation the result that can be found in every physics
textbook. The velocities after a collision are
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_5
39