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Arguments and representative examples are given that suggest that exact energy conservation and Galilei invariance are incompatible in non-relativistic guiding-center mechanics. Provided that this is true in general it also follows that exact energy conservation and Lorentz invariance are incompatible in relativistic guiding-center mechanics. It would furthermore follow that every guiding-center mechanics with exact energy conservation is a non-unique theory owing to the principle of relativity. The paper also presents a Galilei invariant guiding-center mechanics that does not conserve energy.

Energy Conserving Non-relativistic Guiding Center Mechanics and the Galilean Principle of Relativity