Three Particle System
Suppose we have two particles of masses m_{1} and m_{2} already fixed in space at distance r_{12} from each other. Let us bring in a third particle of mass m_{3}, from ∞ to some point P near the first two particles, so that m_{3} finally is at distance r_{13} from m1 and at distance r_{23} from m_{2}.
Now, at any instant, there are two forces acting on m_{3}, viz. the gravitational force F_{31} due to m_{1} and F_{32} due to m_{2}. The total work done in moving m_{3} to point P is given by, Note that the two forces act independently of each other along respective radial directions. That is, for example, we have Note that the two forces act independently of each other along respective radial directions. That is, for example, we have where r and dr in the above integral refer to distances along the radial direction joining particles 1 and 3, at time t. Similarly, we get For conservative forces, the work done is interpreted as the negative change in potential energy. Hence, the increase in gravitational potential energy of the system by joining of third particle is (-W_{3}). The total potential energy of three-particle system becomes,U = U_{12} + ( -W_{3} ) Thus, the total potential energy of the system is the sum of potential energies of each pair of particles taken independently.Remember that ( -W_{3} ) is not the potential energy 'of mass m_{3}'; it is the sum of potential energies of masses (m_{1} and m_{3})and masses (m_{2} and m_{3}).If m_{3} = 1 (unit mass), we define the gravitational field at point P due to masses m_{1} and m_{2} as the net force acting on unit mass at P. where we are now writing r_{1} and r_{2} as the position vectors of point P relative to masses m_{1} and m_{2}. [That is, in fact, r_{1} ≡ r_{31}and r_{2} = r_{32}].Gravitational potential at point P due to masses m_{1} and m_{2} gives the change in potential energy of the system when a unit mass is added to the system at point P. That is, potential ØP at P is the value of ( -W_{3} ) from m_{3} = 1 (unit mass). where r_{1} and r_{2} denote distances of P from m_{1} and m_{2}.