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....?

  • $\begingroup$ @DanYand I'm pretty sure that in field theory, conserved means that the four-divergence vanishes. This only necessarily means that there is a quantity whose time derivative is zero, if the three divergence ('current') vanishes at the boundaries in question. In field theory we tend to talk of 'conserved current' i.e. vanishing four divergence, and not conserved quantity over time. $\endgroup$ – Meep Nov 25 '18 at 15:18
  • $\begingroup$ I am asking for a clarification of what the significance is of a Lagrangian with explicit space/time dependence, considering we often re-define quantities like momentum. So we might say 'conjugate/mechanical momentum is not conserved', the two statements of which are not equivalent, but redefine a quantity which is conserved. $\endgroup$ – Meep Nov 25 '18 at 15:21

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