An important scientific debate took place regarding falling bodies hundreds of years ago, and it still warrants close examination. Galileo argued that in a vacuum all bodies fall at the same rate relative to the earth, independent of their mass. As we shall see, the problem is more subtle than meets the eye -- even in vacuum. In principle the results of a free fall experiment depend on whether falling masses are sequential or concurrent, whether they fall side by side or diametrically opposed. In the current paper we will present both the classical mechanics treatment and the general relativity one. In the case of classical mechanics, we start from the basic equations of motion. On the other hand, the determination of particle equations of motion in gravitational fields in general relativity is done routinely via the use of covariant derivatives. Since the geodesic equations based on covariant derivatives are derived from the Euler-Lagrange equations and since the Euler-Lagrange formalism is very intuitive, easy to derive with no mistakes, there is every reason to use them even for the most complicated situations and this is exactly what we do in the second part of the current paper.
Classical Mechanics, General Relativity, Schwarzschild Metric, Euler-Lagrange Formalism, PACS: 03.30.+p, 52.20.Dq, 52.70.Nc