Among different Lagrangians, null Lagrangians are known for having identically zero the Euler-Lagrange equation and, therefore, they have no effects on the resulting equations of motion. However, there is a special family of null Lagrangians that can be used to convert linear and undriven equations of motion into nonlinear and driven ones. To identify this special family, general null Lagrangians and their gauge functions are constructed for second-order ordinary differential equations of motion describing one-dimensional dynamical systems. The gauge functions corresponding to forces and nonlinearities in a variety of known dynamical systems are presented and novel roles of these functions in dynamics are discussed.

Non-standard Lagrangians are a family of Lagrangians with terms that do not have clearly discernable energy-like forms. A special class among these Lagrangians are non-standard null Lagrangians and their gauge functions, which are used to introduce forces to the laws of dynamics. It is shown that the non-standard Lagrangian for the law of inertia is Galilean invariant but the forces resulting from the non-standard null Lagrangian are not Galilean invariant, except special only time-dependent forces.