Two bodies of masses \(M_{1}\) and \(M_{2}\) are moving in circular orbits of radii \(a_{1}\) and \(a_{2}\) about their centre of mass. The restricted three- body problem concerns the motion of a third small body of mass \(m\left(\ll M_{1}\right.\) or \(M_{2}\) ) in their gravitational field (e.g., a spacecraft in the vicinity of the Earth-Moon system). Assuming that the third body is moving in the plane of the first two, write down the Lagrangian function of the system, using a rotating frame in which \(M_{1}\) and \(M_{2}\) are fixed. Find the equations of motion. (Hint: The identities \(G M_{1}=\omega^{2} a^{2} a_{2}\) and \(G M_{2}=\omega^{2} a^{2} a_{1}\) may be useful, with \(a=a_{1}+a_{2}\) and \(\omega^{2}=G M / a^{3}\).)

Polar coordinates are used to describe the position of the third body: \(r\), the distance from the origin, and \(\theta\), the angle the radial vector makes with the \(x\)-axis. 2. How is the gravitational potential energy expressed in terms of distances and angles? The gravitational potential energy, \(U\), is expressed as: \[ U = -G \frac{M_1 m}{\sqrt{(r\cos\theta-a_1)^2+(r\sin\theta)^2}} - G \frac{M_2 m}{\sqrt{(r\cos\theta+a_2)^2+(r\sin\theta)^2}} \] 3. How is the kinetic energy expressed in the rotating frame? The kinetic energy, \(T\), in the rotating frame is expressed as: \[ T = \frac{1}{2} m (\dot{r}^2 + r^2\dot{\theta}^2) \] 4. What additional energy term is accounted for in this problem due to the rotating frame? The additional energy term accounted for is the fictitious potential energy due to the centrifugal force, \(U_\text{cf}\), which is expressed as: \[ U_\text{cf} = \frac{1}{2} m\omega^2 r^2 \] 5. Write the Lagrangian function in terms of the kinetic and potential energies. The Lagrangian function, \(\mathcal{L}\), is defined as: \[ \mathcal{L} = T - U - U_\text{cf} \] 6. What equations are used to derive the equations of motion for this problem? The Euler-Lagrange equations are used to derive the equations of motion: \[ \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\right) - \frac{\partial \mathcal{L}}{\partial q_i} = 0 \] where \(q_i\) are the generalized coordinates (\(r\) and \(\theta\) in this case).