I What does the total time derivative of a function signify in Noether's Theorem?

[Higgsono]

We can look at infinitesimal transformations in the fields that leaves the Lagrangian invariant, because that implies that the equations of motions are invariant under this transformations. But what really matters is the those transformations that leaves the action invariant. So we can always add a total time derivative of a functions such that the change in the Lagrangian under a infinitesimal transformation is proportional to a total time derivative of a function.

My question is, what is the physical interpretation of this? What does this total time derivative of a function represent?

 

[anorlunda]

What does this total time derivative of a function represent?

You must have some calculus education, so that really can't be your question, can it?

 

[Dale]

So we can always add a total time derivative of a functions such that the change in the Lagrangian under a infinitesimal transformation is proportional to a total time derivative of a function.

That is an interesting idea. I don’t know what such a quantity would represent in general. Perhaps you should calculate the corresponding conserved quantity for a few specific examples to gain insight.