Connections between Symmetries and Conservation Laws
In this presentation, I will discuss several connections between symmetries and conservation laws. In the classical Noether's theorem, if a given system of DEs
has a variational principle, then a continuous symmetry that leaves the action functional invariant to within a divergence yields a conservation law. In applying Noether's theorem, it is natural to use symmetries in evolutionary form. A system of DEs (as written) has a variational principle if and only if its linearized system is self-adjoint.
More generally, a system of DEs has a conservation law if and only if each Euler operator associated with its dependent variables annihilates the scalar product of multipliers with each DE where solutions are replaced by arbitrary functions. The Direct Construction Method yields an integral formula for the corresponding conservation law for any such set of multipliers. On the solution space of the given system of DEs, multipliers are symmetries provided the linearized system is self-adjoint; otherwise they solve the adjoint of the linearized system.
For any system of DEs, a symmetry of the system maps a given conservation law to another conservation law. Moreover, there is a formula to determine the action of the symmetry on the multipliers of the given conservation law and hence one can determine in advance if the resulting conservation law will be new.
Useful conservation laws of a given system of DEs yield equivalent non-locally related systems of DEs. Such non-locally related systems (and some of their subsystems) in turn can yield non-local conservation laws and non-local symmetries of the given system. This leads to the problem of finding complete trees of equivalent non-locally related systems of DEs.