Tractrix with inertia

Abstract

The tractrix is a special type of pursuit trajectory with unphysical properties chosen for its mathematical interest. The trajectory can be made more physically reasonable by including the effects of inertia and of friction. Here the ‘inertial tractrix’ locus is derived analytically using the Euler–Lagrange approach and is found, in the absence of friction, to be a cycloid. The effects of friction are determined by numerical calculation, which show that the tractrix and cycloid loci are limiting cases as the friction coefficient becomes large and small, respectively. The analysis is extended to include variable driving velocity.