Lagrangian equations of motion of particles and photons in a Schwarzschild field

The equations of motion of a particle in the gravitational field of a black hole are considered in a formulation that uses generalized coordinates, velocities, and accelerations and is convenient for finding the integrals of motion. The equations are rewritten in terms of the physical velocities and accelerations measured in the Schwarzschild frame by a stationary observer using proper local length and time standards. The attractive force due to the field and the centripetal acceleration of a particle is proportional to the particle kinetic energy $m/\sqrt{1 - v^2}$, consistently with the fact that the particle kinetic energy and the photon energy $\hbar\omega$ in the field increase by the same factor compared with their values without a field. The attraction exerted on particles and photons by a gravitational field source is proportional to their kinetic energies. The particle trajectory in the ultrarelativistic limit $v \to 1$ coincides with the photon trajectory.