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We show that if a Lagrangian is invariant under a transformation (with theinvariance defined in the standard manner), then the equations of motionobtained from it maintain their form under the transformation. We also showthat the converse is not true, giving examples of equations of motion that areform-invariant under a transformation, but these equations can be derived froma Lagrangian that is not invariant under such transformation. The conclusionsare valid for discrete or continuous systems.

Symmetries of the equations of motion that are not shared by the Lagrangian