This chapter investigates some aspects of the relation of governing equations, space-time symmetries, and fundamental balances in non-relativistic field theories. Variational principle construction methods for a given differential equation are emphasized. The chapter explains that symmetry has nothing to do with coordinates. It is frequently said that the space-time symmetries mean that the origin of the coordinate system and the direction of the coordinate axis can be chosen arbitrarily. This has to be replaced with the assertion that space-time symmetry means: if the system is translated and rotated in space-time (a space-time isomorphism is applied to it) then the same (or an equivalent) system is achieved. The possibility to consider symmetry principles with different variational construction methods is also discussed. The method of variable transformations is treated using the example of Maxwell equations. Finally, the chapter concludes that the requirements of physically meaningful Noether balances can fix the freedom of the construction (the arbitrary function f), and this freedom can help to develop the existing balances to treat more general problems, for example, to find variational principles for the Maxwell equations in materials with nonlinear polarization.
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