  We propose the Lagrangian formulation for describing the motion of a test
particle in a general relativistic, stationary, and axially symmetric
spacetime. The test particle is also affected by a radiation field, modeled as
a coherent flux of photons traveling along the null geodesics of the background
spacetime, including the general relativistic Poynting-Robertson effect. The
innovative part of this work is to prove the existence of the potential linked
to the dissipative action caused by the Poynting-Robertson effect in General
Relativity through the help of an integrating factor, depending on the energy
of the system. Generally such kinds of inverse problems involving dissipative
effects might not admit a Lagrangian formulation, especially in General
Relativity there are no examples of such attempts in literature so far. We
reduce this general relativistic Lagrangian formulation to the classic case in
the weak field limit. This approach facilitates further studies in improving
the treatment of the radiation field and it contains for example some
implications for a deeper comprehension of the gravitational waves.
