LETTERS AND COMMENTS Inertial forces and gauge invariance

An elementary mechanical example is discussed in which the appearance of the easiest inertial force leads naturally to two different but equivalent Lagrangians. This provides a family of simple examples to discuss gauge invariance in analytical mechanics. The physical meaning of the gauge in these examples is also analysed. The equivalence of two Lagrangians differing in a total time derivative is mentioned in many texts about analytical dynamics [1, 2] where it is often proposed as an exercise [3], its proof being elementary. Despite the paramount importance of this result in the Lagrangian formulation of field theories it goes unnoticed by most students of classical mechanics. This may be due to the fact that in courses on analytical mechanics it is only applied (if at all) to the gauge invariance of electrodynamics [3], which is of course its most interesting elementary application but this may be difficult to be fully understood by a student with little background on electromagnetism. The purpose of this note is to discuss a very simple mechanical example in which two natural approaches lead to Lagrangians differing by a gauge transformation