A modified model of coupled pendulums

We consider a modified system of two pendulums rods of which intersect and slide without any friction. The pendulums are connected by an elastic linear spring and arranged in a fixed vertical plane of the uniform gravity field. We have shown that there are symmetric and asymmetric equilibrium solutions with respect to the vertical axis. It is revealed that the stability of the model depends on two parameters, the first one specifies the spring stiffness, and the second one defines the distance between the hinges. The conditions of stability and instability of the symmetric equilibrium are obtained in the upper and lower position of pendulums. The analysis of asymmetric equilibrium solutions and stability conditions is carried out for long pendulums. Comparison with the sympathetic pendulums model proposed by Sommerfeld indicates that asymmetric solutions exist only for the modified model.