50
6 Lattice and the Continuum
For every point of time t in our one-dimensional chain, the single force F i+1 i that
acts on the mass x i to the right, is per definition equal to the tension τ (x i , t). All of
the forces are directed at the particle’s line of junction, in the x-direction, and can
therefore simply be added like numbers (their vector properties do not apply here).
Therefore, F i−1 i = −F i i−1 = −τ (x i−1 , t) and the following are valid,
F i−1 i + F i+1 i = −F i i−1 + F i+1 i = −τ (x i−1 , t) + τ (x i , t) .
(Only in the one-dimensional case tension is also a force; in a three dimensional
continuum, tension is a force/area, see Chap. 23). The equally distributed positions
of masses x i =
L
N
i are, in the passing to the limit assimilated into the continuous
variable x, so that
x i =
L
N
i −→ x , x i+1 −→ x + x
and
F i−1 i + F i+1 i −→ τ (x − x, t) + τ (x, t) .
Using the Taylor series expansion for τ with a sufficiently small x, this becomes
F i−1 i + F i+1 i −→
τ (x, t) −
∂τ (x, t)
∂x
x
+ τ (x, t) =
∂τ (x, t)
∂x
x . (64)
We replace the single masses with a continuous mass density ρ and the timedependent displacements q i (t) of the single masses with the function s(x, t),
m i −→ ρx , q i (t) −→ s(x, t) .
(65)
For the passing to the limit from the Newtonian equations for single masses to a
continuum, we place
d
dt
m i
d
dt
q i
−→ x ρ
∂
∂t
∂s(x, t)
∂t
,
k =i
F ki −→
∂τ (x, t)
∂x
x .
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(66)
The first passing to the limit (66) contains an important approximation that is somewhat complicated. The Newtonian inertia term d P/dt =
d
dt
((m v) describes the
change with respect to time of a momentum carried by a certain moving mass m.
The volume V m (in the one-dimensional case x m ) occupied by this mass also
changes. During the transition to a continuum, whose motion we will describe in
the scope of a field theory, we are interested in a completely other physical quantity. This quantity is the change with respect to time of momentum contained in a
6 Lattice and the Continuum
For every point of time t in our one-dimensional chain, the single force F i+1 i that
acts on the mass x i to the right, is per definition equal to the tension τ (x i , t). All of
the forces are directed at the particle’s line of junction, in the x-direction, and can
therefore simply be added like numbers (their vector properties do not apply here).
Therefore, F i−1 i = −F i i−1 = −τ (x i−1 , t) and the following are valid,
F i−1 i + F i+1 i = −F i i−1 + F i+1 i = −τ (x i−1 , t) + τ (x i , t) .
(Only in the one-dimensional case tension is also a force; in a three dimensional
continuum, tension is a force/area, see Chap. 23). The equally distributed positions
of masses x i =
L
N
i are, in the passing to the limit assimilated into the continuous
variable x, so that
x i =
L
N
i −→ x , x i+1 −→ x + x
and
F i−1 i + F i+1 i −→ τ (x − x, t) + τ (x, t) .
Using the Taylor series expansion for τ with a sufficiently small x, this becomes
F i−1 i + F i+1 i −→
τ (x, t) −
∂τ (x, t)
∂x
x
+ τ (x, t) =
∂τ (x, t)
∂x
x . (64)
We replace the single masses with a continuous mass density ρ and the timedependent displacements q i (t) of the single masses with the function s(x, t),
m i −→ ρx , q i (t) −→ s(x, t) .
(65)
For the passing to the limit from the Newtonian equations for single masses to a
continuum, we place
d
dt
m i
d
dt
q i
−→ x ρ
∂
∂t
∂s(x, t)
∂t
,
k =i
F ki −→
∂τ (x, t)
∂x
x .
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(66)
The first passing to the limit (66) contains an important approximation that is somewhat complicated. The Newtonian inertia term d P/dt =
d
dt
((m v) describes the
change with respect to time of a momentum carried by a certain moving mass m.
The volume V m (in the one-dimensional case x m ) occupied by this mass also
changes. During the transition to a continuum, whose motion we will describe in
the scope of a field theory, we are interested in a completely other physical quantity. This quantity is the change with respect to time of momentum contained in a