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Null Lagrangians and their gauge functions are derived for given standard andnon-standard Lagrangians. The obtained standard null Lagrangians generalizethose previously found but the non-standard null Lagrangians are new. The gaugefunctions are used to make the action invariant and introduce the exact nullLagrangians, which form a new family of null Lagrangians. The conditionsrequired for the action to be invariant are derived for all null Lagrangianspresented in this paper, thus, making them exact. As a specific application,the exact null Lagrangians are derived for a second-order ordinary differentialequation. The application shows that the exact null Lagrangians can be obtainedfor any ordinary differential equation with known Lagrangian.

Null Lagrangians and Invariance of Action: Exact Gauge Functions