JGP - Vol.7. n. 2, 1990 A study of the action from kinematical integral geometry point of view M. A. DEL OLMO AND M. SANTANDER Departamento de Fisica Teórica. Facultad de Ciencias Universidad de Valladoljd 47011 - VALLADOLID, SPAIN Abstract. We develop here an interpretation of the classical nonrelativistic and rela- tivistic action for a point particle as related to geometric measures of sets of straight lines (inertial motions) associated in anatural way to closed timeike circuits in space- time. This allows a point of view for the action common to classical and relativistic mechanics. Fw-thennore the results are not restricted to the free case and also holds for parti des in some potentials (homogeneous field and the harmonic oscillator). 1. INTRODUCTION Action is the single most important quantity in physics [1] and it is worth to explore any path that could rise new points of view about it. That is the aim of this paper. The relativistic action for a free particle has a well-known geometrical meaning, as it is proportional to the length of (proper time along) the particle worldline. But in the non-relativistic case, the length of a worldline (the lapse of universal time), is path inde- pendent. Action is therefore introduced in classical mechanics as a path dependent quan- tity, without any known geometrical meaning. The relationship between both actions is described in geometric terms by a <<timelike>> contraction from Minkowski space-time: in the proximity of a timelike straight line, the proper time along a wordline with fixed Key-Words: Inte,gral Geometry, Action, Kinematics. AMS: Classification Numbers: 53C65, 70B05, 70D99, 53C80.