Chapter 6
Lattice and the Continuum
In the last chapter,we conducted a step-by-step transition from a point lattice to a
continuum. We showed starting from a linear chain of elastic interacting single masses
how to arrive at an oscillating rod. All we have to do to achieve this is to continuously
halve the masses and the connecting elastic springs. The mathematical result at the
end of this process, from a coupled system of ordinary differential equations for
single masses (31), is the wave equation for an elastic rod (61). One says of such
a constructed continuum, that its internal forces are based on contact interaction
or on the continuous action. In this connection, we talk about the hypothesis of
continuous action. We would say in reference to single masses that every mass of
the N -particle system only interacts with its direct neighbours. This factual situation
should be maintained in the passing to the limit N −→ ∞. We will now show
how the transition from normal differential equations for single masses m i to the
partial differential equations for a continuum of the mass density ρ can easily be
described due to the continuous action hypothesis. We will once again consider the
one-dimensional case, where we distinguish two situations, for which we however,
at the end, receive one and the same result.
(a) The strict one-dimensional problem, in other words longitudinal oscillations:
Up to now, we have only taken this case into consideration and observed the longitudinal oscillations of a rod. The displacements s = s(x, t) were made in the direction
of the rod, which we positioned on the x-axis. The Newtonian equations (57) are
valid for the single masses m i . For reasons of simplicity, we ignored the external
forces F a (in the passing to the limit of a continual mass distribution the external
forces act as a force field F(x)). The total sum of the interaction forces refers only
to its direct neighbours due to the hypothesis of continuous action, so that
d
dt
m i
d
dt
q i
=
k =i
F ki = F i−1 i + F i+1 i ,
F ik = −F ki .
⎫
⎪ ⎬
⎪ ⎭
(63)
© 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_6
49
Lattice and the Continuum
In the last chapter,we conducted a step-by-step transition from a point lattice to a
continuum. We showed starting from a linear chain of elastic interacting single masses
how to arrive at an oscillating rod. All we have to do to achieve this is to continuously
halve the masses and the connecting elastic springs. The mathematical result at the
end of this process, from a coupled system of ordinary differential equations for
single masses (31), is the wave equation for an elastic rod (61). One says of such
a constructed continuum, that its internal forces are based on contact interaction
or on the continuous action. In this connection, we talk about the hypothesis of
continuous action. We would say in reference to single masses that every mass of
the N -particle system only interacts with its direct neighbours. This factual situation
should be maintained in the passing to the limit N −→ ∞. We will now show
how the transition from normal differential equations for single masses m i to the
partial differential equations for a continuum of the mass density ρ can easily be
described due to the continuous action hypothesis. We will once again consider the
one-dimensional case, where we distinguish two situations, for which we however,
at the end, receive one and the same result.
(a) The strict one-dimensional problem, in other words longitudinal oscillations:
Up to now, we have only taken this case into consideration and observed the longitudinal oscillations of a rod. The displacements s = s(x, t) were made in the direction
of the rod, which we positioned on the x-axis. The Newtonian equations (57) are
valid for the single masses m i . For reasons of simplicity, we ignored the external
forces F a (in the passing to the limit of a continual mass distribution the external
forces act as a force field F(x)). The total sum of the interaction forces refers only
to its direct neighbours due to the hypothesis of continuous action, so that
d
dt
m i
d
dt
q i
=
k =i
F ki = F i−1 i + F i+1 i ,
F ik = −F ki .
⎫
⎪ ⎬
⎪ ⎭
(63)
© 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_6
49