Suppose that we have a dynamical system specified in terms of
independent variables
,
dependent variables
, and a Lagrangian function
of some finite order
. Here
is the collection of all
th order partial derivatives of the dependent variables. As a general rule, latin indices
from the middle of the alphabet take the values
, greek indices take the values
, and the summation convention apply to them. Multiindex notation for the latin indices is also introduced as follows. A multiindex
of length
is an ordered list
of
ordinary indices. The length is denoted as
. The summation convention does not directly apply to multiindices since the summation over lengths needs to be displayed explicitly, e.g.
The variation of the Lagrangian with respect to an arbitrary variation
of the dependent variables is
and applying the inverse product rule of differentiation we get
where
are the Euler-Lagrange expressions of the Lagrangian, and the coefficients
(Lagrangian momenta) are given by
Variational symmetries
A variation
is an infinitesimal symmetry of the Lagrangian
if
under this variation. It is an infinitesimal quasi-symmetry if there is a current
such that
.
It should be remarked that it is possible to extend infinitesimal (quasi-)symmetries by including variations with
as well, i.e. the independent variables are also varied. However such symmetries can always be rewritten so that they act only on the dependent variables. Therefore, in the sequel we restrict to so-called vertical variations where
.
For Noether's second theorem, we consider those variational symmetries (called gauge symmetries) which are parametrized linearly by a set of arbitrary functions and their derivatives. These variations have the generic form
where the coefficients
can depend on the independent and dependent variables as well as the derivatives of the latter up to some finite order, the
are arbitrarily specifiable functions of the independent variables, and the latin indices
take the values
, where
is some positive integer.
For these variations to be (exact, i.e. not quasi-) gauge symmetries of the Lagrangian, it is necessary that
for all possible choices of the functions
. If the variations are quasi-symmetries, it is then necessary that the current also depends linearly and differentially on the arbitrary functions, i.e. then
, where
For simplicity, we will assume that all gauge symmetries are exact symmetries, but the general case is handled similarly.
Noether's second theorem
The statement of Noether's second theorem is that whenever given a Lagrangian
as above that admits gauge symmetries
parametrized linearly by
arbitrary functions and their derivatives, then there exist
linear differential relations between the Euler-Lagrange equations of
.
Combining the first variation formula together with the fact that the variations
are symmetries, we get
where on the first term proportional to the Euler-Lagrange expressions, further integrations by parts can be performed as
where
in particular for
,
Hence, we have an off-shell relation
where
with
. This relation is valid for any choice of the gauge parameters
. Choosing them to be compactly supported, and integrating the relation over the manifold of independent variables, the integral total divergence terms vanishes due to Stokes' theorem. Then from the fundamental lemma of the calculus of variations, we obtain that
identically as off-shell relations (in fact, since the
are linear in the Euler-Lagrange expressions, they necessarily vanish on-shell). Inserting this back into the initial equation, we also obtain the off-shell conservation law
.
The expressions
are differential in the Euler-Lagrange expressions, specifically we have
where
Hence, the equations
are
differential relations to which the Euler-Lagrange expressions are subject to, and therefore the Euler-Lagrange equations of the system are not independent.
Converse result
A converse of the second Noether them can also be established. Specifically, suppose that the Euler-Lagrange expressions
of the system are subject to
differential relations
Letting
be an arbitrary
-tuple of functions, the formal adjoint of the operator
acts on these functions through the formula
which defines the adjoint operator
uniquely. The coefficients of the adjoint operator are obtained through integration by parts as before, specifically
where
Then the definition of the adjoint operator together with the relations
state that for each
-tuple of functions
, the value of the adjoint on the functions when contracted with the Euler-Lagrange expressions is a total divergence, viz.
therefore if we define the variations
the variation
of the Lagrangian is a total divergence, hence the variations
are quasi-symmetries for every value of the functions
.