Problem 15 - Classical Mechanics

Question

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 \(\omega^{2}=G M / a^{3}\).)

Step-by-Step Solution

Answer

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).

Step 1: Write down the positions of the masses in the rotating frame

In the rotating frame, \(M_1\) and \(M_2\) are fixed at \((a_1, 0)\) and \((-a_2, 0)\), respectively. The position of mass \(m\) is given by the polar coordinates (\(r\), \(\theta\)) in the same plane.

Step 2: Find the gravitational potential energy of m

The gravitational potential energy \(U\) of mass \(m\) due to \(M_1\) and \(M_2\) is given by: \[ U = -G \frac{M_1 m}{\sqrt{(r\cos\theta-a_1)^2+(r\sin\theta)^2}} - G \frac{M_2 m}{\sqrt{(r\cos\theta+a_2)^2+(r\sin\theta)^2}} \]

Step 3: Find the kinetic energy of m in the rotating frame

The kinetic energy \(T\) of mass \(m\) in polar coordinates can be expressed as: \[ T = \frac{1}{2} m (\dot{r}^2 + r^2\dot{\theta}^2) \]

Step 4: Account for the centrifugal force in the rotating frame

The centrifugal force introduces a fictitious potential energy (due to a fictitious force) to the system: \[ U_\text{cf} = \frac{1}{2} m\omega^2 r^2 \]

Step 5: Write down the Lagrangian function

The Lagrangian function \(\mathcal{L}\) is defined as the difference between the kinetic and potential energies: \[ \mathcal{L} = T - U - U_\text{cf} \] Substituting the expressions for \(T, U\) and \(U_\text{cf}\), we get: \[ \mathcal{L} = \frac{1}{2} m (\dot{r}^2 + r^2\dot{\theta}^2) + G\left(\frac{M_1 m}{\sqrt{(r\cos\theta-a_1)^2+(r\sin\theta)^2}} + \frac{M_2 m}{\sqrt{(r\cos\theta+a_2)^2+(r\sin\theta)^2}}\right) - \frac{1}{2} m\omega^2 r^2 \]

Step 6: Find the equations of motion

We can use the Euler-Lagrange equations to derive the equations of motion for the small mass \(m\). They can be written as: \[ \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\right) - \frac{\partial \mathcal{L}}{\partial q_i} = 0 \] where \(q_i\) are the generalized coordinates (in this case, \(r\) and \(\theta\)). Computing the required derivatives and applying the Euler-Lagrange equations, we'll obtain the equations of motion.

Key Concepts

Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, which provides a powerful method for analyzing the dynamics of a system, especially in complex situations. It's particularly useful in dealing with the restricted three-body problem, as it allows us to systematically account for the interplay of kinetic and potential energies and other forces acting on the system.

In this framework, the dynamics of a system are described by the Lagrangian function, denoted as \(\mathcal{L}\). The function is expressed as the difference between the kinetic energy \(T\) and the potential energy \(U\) of the system:

• \(\mathcal{L} = T - U\)

This function encodes all the necessary information about the energy distribution and constraints of the system.

For our three-body problem, the Lagrangian function must also incorporate the effects of the rotating frame and fictitious forces, such as the centrifugal force.

Equations of motion

The equations of motion in the Lagrangian framework are derived using the Euler-Lagrange equations. These equations tell us how the generalized coordinates, like position and angle, evolve over time, based on the system's Lagrangian.

The Euler-Lagrange equations are given by:

• \(\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}_i}\right) - \frac{\partial \mathcal{L}}{\partial q_i} = 0\)

Here, \(q_i\) represents the generalized coordinates, and \(\dot{q}_i\) their time derivatives (velocities).

In the context of the three-body problem, these equations will help us determine the motion of the small mass \(m\) under the gravitational influence of the larger masses \(M_1\) and \(M_2\). The systematic approach of using Lagrangian mechanics allows for clean, comprehensive derivations of these complex dynamical equations.

Rotating frames

A rotating frame is a non-inertial frame of reference that rotates in relation to an inertial frame. In our exercise, we're analyzing the three-body problem in a frame where the massive bodies \(M_1\) and \(M_2\) remain fixed, simplifying the system analysis.

While a rotating frame simplifies certain analyses, it introduces fictitious forces like the centrifugal force, which must be accounted for in the Lagrangian function. Fictitious forces do not arise from any physical interaction but from the acceleration of the frame itself.

To adjust for these forces, we introduce additional potential energy terms in our Lagrangian. This ensures that the dynamics account for the effects of these added forces, allowing us to remain within the framework of Lagrangian mechanics efficiently. Understanding the relationship between rotating frames and non-inertial forces is crucial for dealing with complex systems, like those seen in celestial mechanics.

Gravitational potential energy

Gravitational potential energy in a system arises from the gravitational attraction between masses. It plays a key role in the dynamics of any gravitationally bound system.

For the small mass \(m\) in our exercise, gravitational potential energy \(U\) is determined by its distance from the two larger masses \(M_1\) and \(M_2\).

The potential energy can be expressed as:

• \(U = -G \frac{M_1 m}{\sqrt{(r\cos\theta-a_1)^2+(r\sin\theta)^2}} - G \frac{M_2 m}{\sqrt{(r\cos\theta+a_2)^2+(r\sin\theta)^2}}\)

Here, \(G\) is the gravitational constant, and the terms inside the square root quantify the distances between \(m\) and \(M_1\), and between \(m\) and \(M_2\).

This expression allows us to incorporate gravitational interactions into the Lagrangian, thus enabling a comprehensive analysis of the small mass's motion under the influence of both larger bodies.