First off, I have seen this post here which asks seems to ask my question, but it is not properly answered. If the Lagrangian has explicit time dependence, then the total energy, and Hamiltonian, is not conserved. So I presume explicit spatial dependence means momentum is not conserved?
Can we then modify our definition of energy and momentum such that we do have conserved quantities? For example, the canonical momentum for a particle in electromagnetic field (not considering field theory here) has a term with yhe magnetic vector potential and is not simply the mechanical momentum. Does a similar thing apply for field theory? I am getting slightly confused as to what are conserved quantities up to a definition and what isn't. It seems to me that whenever we have a term in the Lagrangian that would break a symmetry and seemingly prevent us from saying that 'quantity $x$ is conserved', we can couple the Lagrangian to another field and redefine quantity $x$ such that $x$ is not conserved....?