In the simple example of a stationary electric field (and some other quantum mechanical examples) it is shown in the papers

https://arxiv.org/pdf/physics/0506203.pdf https://arxiv.org/pdf/1302.1212.pdf

that a particular gauge transformation can lead to different Hamiltonian's and Lagrangian's for what are established as closed systems which then lead to different physical interpretations. In one calculation, energy is conserved due to a time-independent Hamiltonian and with the addition of a time dependent gauge transformation, the Hamiltonian becomes time-dependent which means energy is not conserved.

My question is what is the significance of this result in terms of the changes a particular gauge can make on the Hamiltonian of a system which may leads to nonsensical physical interpretations? Although now learning about this area, it seems to me that a particular gauges ability to change a conserved quantity of a system is negative thing. Thus, in the grand scheme of things, I would ask what the resolution of this seeming paradox is, why it comes about, the meaning behind the existence of these paradoxes, and the relation between the Hamiltonian and Lagrangian formalism and gauge transformations.

Mathematical details are welcome and if there any any sources I should read up on myself I would appreciate such suggestions. Thank you in advance!


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