Construction of Lagrangian and Hamiltonian structures starting from one constant of motion

The problem of the construction of Lagrangian and Hamiltonian structures starting from two first-order equations of motion is presented. This approach requires the knowledge of one (time independent) constant of motion for the dynamical system only. The Hamiltonian and Lagrangian structures are constructed, the Hamilton–Jacobi equation is then written and solved, and the second (time dependent) constant of the motion for the problem is explicitly exhibited.