The equilibrium positions of solid bodies on an elastic wire with energy involving arc length

In this paper, we study the equilibrium positions of two different solid bodies, namely, a square and a two-disk, allowing to move freely on an elastic wire. The energy under consideration is involving the arc length of admissible function. We obtain the necessary and sufficient condition for the existence of minimizers on the weight of the solid body in each case. We also characterize the position(s) of the minimizer(s) in each case.