22 Particles and Fields
243
potential energy. Correspondingly, the Lagrangian density L is in the field theory
identical to the difference between the potential and kinetic energy densities, T − W .
The Lagrangian density L for the field q = q(x, t) of the sine-Gordon equation
is, according to Rubinstein [84]
L = L
q ,
∂q
∂x
,
∂q
∂t
= −
α
2
∂q
∂x
∂q
∂x
−
1
c 2
o
∂q
∂t
∂q
∂t
+ A
cos
2π
a
q
− 1
.
(283)
The parameters α and A guarantee that we are dealing with the Lagrangian density
of the sine-Gordon equation for a dislocation inside of a crystal; see below. Where
do the individual terms of this Lagrangian density come from?
In order to find this out, we need to calculate the contributions to the potential and
kinetic energy densities of the field q.
Firstly the potential energy density: The relative strain ε = ∂q/∂x generates,
according to our Eq. (86), the stress τ of the linear chain on the basis of the modulus
of elasticity of this chain, which we will designate, in order to prevent mistakes in
identity with the particle energy E calculated according to (282), as the line tension
σ, which it is identical with (see Chaps. 6 and 8), thus
τ = σ ε = σ
∂q
∂x
.
This stress τ is, as we know, in the one-dimensional case simply a force acting
at the position x. We wish to calculate the work W 1 x needed to increase the
displacement q(x, t) by the displacement q, which is also dependent on its location,
between the position x and x + x with an existing stress state τ = τ (x, t). The
quantity W 1 is the increase of energy density on the length x caused by q. On
the piece between x and x + x, we approximately describe the initial state q using
the first term of its Taylor series,
q = q(x, t) +
∂q
∂t
x .
A displacement q(x, t) at the left-hand side of x changes the localised energy
W 1 x on the piece x by −σ
∂q
∂x
q(x, t). The displacement at the right-hand
side brings a change of +σ
∂q
∂x
q(x + x, t). All together, this would bring,
W 1 x = −σ
∂q
∂x
q(x, t) + σ
∂q
∂x
q(x + x, t) ,
W 1 x = −σ
∂q
∂x
q(x, t) + σ
∂q
∂x
q(x, t) + σ
∂q
∂x
∂
∂x
q(x, t))x .
243
potential energy. Correspondingly, the Lagrangian density L is in the field theory
identical to the difference between the potential and kinetic energy densities, T − W .
The Lagrangian density L for the field q = q(x, t) of the sine-Gordon equation
is, according to Rubinstein [84]
L = L
q ,
∂q
∂x
,
∂q
∂t
= −
α
2
∂q
∂x
∂q
∂x
−
1
c 2
o
∂q
∂t
∂q
∂t
+ A
cos
2π
a
q
− 1
.
(283)
The parameters α and A guarantee that we are dealing with the Lagrangian density
of the sine-Gordon equation for a dislocation inside of a crystal; see below. Where
do the individual terms of this Lagrangian density come from?
In order to find this out, we need to calculate the contributions to the potential and
kinetic energy densities of the field q.
Firstly the potential energy density: The relative strain ε = ∂q/∂x generates,
according to our Eq. (86), the stress τ of the linear chain on the basis of the modulus
of elasticity of this chain, which we will designate, in order to prevent mistakes in
identity with the particle energy E calculated according to (282), as the line tension
σ, which it is identical with (see Chaps. 6 and 8), thus
τ = σ ε = σ
∂q
∂x
.
This stress τ is, as we know, in the one-dimensional case simply a force acting
at the position x. We wish to calculate the work W 1 x needed to increase the
displacement q(x, t) by the displacement q, which is also dependent on its location,
between the position x and x + x with an existing stress state τ = τ (x, t). The
quantity W 1 is the increase of energy density on the length x caused by q. On
the piece between x and x + x, we approximately describe the initial state q using
the first term of its Taylor series,
q = q(x, t) +
∂q
∂t
x .
A displacement q(x, t) at the left-hand side of x changes the localised energy
W 1 x on the piece x by −σ
∂q
∂x
q(x, t). The displacement at the right-hand
side brings a change of +σ
∂q
∂x
q(x + x, t). All together, this would bring,
W 1 x = −σ
∂q
∂x
q(x, t) + σ
∂q
∂x
q(x + x, t) ,
W 1 x = −σ
∂q
∂x
q(x, t) + σ
∂q
∂x
q(x, t) + σ
∂q
∂x
∂
∂x
q(x, t))x .