Lagrangians and Hamiltonians for one-dimensional autonomous systems

An equation is obtained to find the Lagrangian for a one-dimensional autonomous system. The continuity of the first derivative of its constant of motion is assumed. This equation is solved for a generic nonconservative autonomous system that has certain quasi-relativistic properties. A new method based on a Taylor series expansion is used to obtain the associated Hamiltonian for this system. These results have the usual expression for a conservative system when the dissipation parameter goes to zero. An example of this approach is given.