Consider a gravitational three-body system with the masses m1, m2 and m3 such that each mass of a three-body system experiences the force of gravitational attraction from the other members of the system. The free-body diagram for the three-body system is given in the Figure below where G represents the center of mass for the system. As a three-body system, you may, for example, consider the Earth, the Moon and the spacecraft system Z R R3 = Y - 142 R - R 1 R - R R 441 R3-R R2 m1 F12 F21 G m2 The equations of motion for the three-body system are given in Appendix C of Orbital Mechanics for Engineering Students as R (R - R)- k=1 3 (R - R). (R3 - R) F13 Inertial frame 3 Fik = (-1/2 (Tik - Tj.k) Tjk)), 1 F23 F32- F31 H3 R - R3 _mi-" i,k i=1 H3 R - R3 R3 142 |R3 - R| where the gravitational parameter , is defined as , Gm. a) Show that the Lagrangian of the system can be written as 3 3 3 - (-). L: 2 Tij m3 (R - R), (R - R3) (R3-R), X where the index i counts the masses, the index j counts the masses other than the th mass, k counts the Cartesian coordinates as, for example, for the coordinates (1, 3, 21) of the first mass x1,1 = x1, 71,2 = y and 21,3 = 2, and rij is defined rij = =1 (, k Xj.k.) . b) Using the Lagrange equations of motion, show that the equations of motion can be found as i = 1, 2, 3, k = 1,2,3, where the index i counts the masses, the index j counts the masses other than the ith mass, and the k index counts the Cartesian coordinates k = 1,2,3.