The extensible Kapitza pendulum: some considerations on a classic stability problem

Some plain considerations are provided on the influence of axial deformation on the stability of the upper equilibrium position of the Kapitza pendulum with respect to the linearisation or non-linearisation of the associated Lagrange’s equations. Following a very uncomplicated approach and fully accounting for the non-linearity of the problem, it is shown that in the case of the extensible Kapitza pendulum the dynamical behaviour of the system cannot be always correctly captured by a simple linearisation about the upper equilibrium point and a phenomenon related to the degree of approximation can take place for this dynamic system that replicates what happens in the case of the stability of equilibrium of simple axially extensible systems. Also, it is remarked that the introduction of axial deformation may play the same role as the addition of damping.