An illustrative application of the Lie symmetries in the context of first-order mechanical systems: Hathaway’s circular pursuit problem

This work illustrates how one could apply Lie point symmetries for finding the analytical solution to first-order mechanical systems. Although the classical Lie method constitutes a powerful tool for solving differential equations, an obstacle appears in the case of systems of first-order equations because they admit an infinite number of symmetries, and it is not possible to compute them by following a systematic procedure. To overcome this difficulty, we follow the idea exposed in Nucci (J. Math. Phys. 37: 1772–1775, 1996), Nucci (Electr. J. Diff. Eqn. 12: 87–101, 2005), Nucci (J. Math. Phys. 42: 746, 2001) consisting of transforming the original system to an equivalent system in which one of the equations is of second–order. The presented approach is applied to Hathaway’s circular pursuit problem, leading to the analytical solution of the system expressed in terms of the general solution to an Abel equation.