he reference for most of material here is
The main result of this chapter is negative: even though there is a Lagrangian
for the heat equation nevertheless the Hamiltonian technique does not apply.
However, the method of characteristics is separately useful.
We are considering a PDE of the
is an unknown function
refers to the vector of derivatives
and the function
is smooth. We will use the following notation for the function
(First order PDE)