Consider the following general functional ,
where is a known Lagrangian density,
is an unknown function,
and and are its first-order derivatives:
Then the calculus of variations
states that the which minimizes or maximizes
can be found by solving this Euler-Lagrange equation:
Now, the first steps are similar to the derivation of the
(which is a special case of Noether’s theorem):
we need to find relations between the explicit dependence
of on the free variables ,
and its implicit dependence via , and .
Let us start with .
The “hard” (explicit + implicit) derivative
is given by the chain rule:
Inserting the Euler-Lagrange equation into the first term
and using that :
Leading to the following expression for the “soft” (explicit only) derivative :
And then by going through the same process for the other variable ,
we arrive at:
Now we define the so-called stress-energy tensor as a useful abbreviation
(the name comes from its application in relativity),
where is the Kronecker delta:
Such that the two relations we just found can be written as follows:
And with this definition of
we can also rewrite the Euler-Lagrange equation in the same way,
noting that :
These three equations are the framework we need.
What happens if does not explicitly contain , so is zero?
Then the corresponding equation clearly turns into:
Such continuity relations are very common in physics.
This one effectively says that if increases with ,
then must decrease with by a certain amount.
Yes, this is very abstract, but when you apply this technique
to a specific physical problem, and
are usually quantities with a clear physical interpretation.
we get analogous continuity relations,
so there seems to be a pattern here:
if has a continuous symmetry
(i.e. there is a continuous transformation
with no effect on the value of ),
then there exists a continuity relation specific to that symmetry.
This is the qualitative version of Noether’s theorem.
In general, for ,
a continuous transformation (not necessarily a symmetry)
consists of shifting the coordinates as follows:
Where is the amount of shift,
and are parameters
controlling the direction of the shift in -space.
Given a specific , suppose we have found a continuous symmetry,
i.e. a direction
such that the value of is unchanged by the shift, meaning:
Where we set to get rid of it.
Negating and inserting our three equations yields:
This is a continuity relation!
Let us make this clearer by defining some current densities:
So that the above equation can be written
in the standard form of a continuity relation:
This is the quantitative version of Noether’s theorem:
for every symmetry we can find,
Noether gives us the corresponding continuity relation.
This result is easily generalized to more variables
and/or more unknown functions .
Continuity relations tell us about conserved quantities.
Of the free variables ,
we choose one as the dynamic coordinate (usually )
and then all others are transverse coordinates.
Let us integrate the continuity relation over all transverse variables:
Usually the problem’s boundary conditions ensure that ,
in which case is a conserved quantity (i.e. a constant)
with respect to the dynamic coordinate .
In the 1D case
(i.e. if is a Lagrangian rather than a Lagrangian density),
the current density does not exist,
so the conservation of the current is clearly seen:
- O. Bang,
Nonlinear mathematical physics: lecture notes, 2020,