Two masses in relative Hooke’s potential and elliptic integrals

We consider the relative motion of the system of two masses connected by a spring. We analyze it in a range of the Hooke’s law and show that the equations of the relative motion of the system are nonlinear once the equilibrium length of the spring is nonzero. Although the way of deriving the equations of motion is standard in classical mechanics solving them is a complicated and interesting problem of mathematical physics. The analysis leads naturally to elliptic integrals. We obtain complete formulas in an interesting, from both mathematical and physical point of view, way. Our analysis might be useful in some problems of molecular dynamics of diatomic molecules.