Standard and non-standard Lagrangians that give the same equation of motion
are significantly different in their forms, as the latter do not have terms
that clearly discernable energy-like expressions. A special family of these
Lagrangians are null Lagrangians and their gauge functions. It is shown that
non-standard null Lagrangians and their gauge functions can be used to
introduce dissipative forces to dynamical systems. With standard null
Lagrangians being known for introducing non-dissipative forces, the presented
results allow for a complete picture of novel roles played by null Lagrangians
in introducing forces to dynamics. Applications of the results to Newton’s laws
are presented and their Galilean invariance is discussed.
