Question

Although the main topic of this chapter is the motion of two particles subject to no external forces, many of the ideas (for example, the splitting of the Lagrangian L into two independent pieces L = Lcm + Lrel as in Equation (8.13)] extend easily to more general situations. To illustrate this, consider the following: Two masses my and my move in a uniform gravitational field g and interact Problems for Chapter 8 321 via a potential energy U(r).

(a) Show that the Lagrangian can be decomposed as in (8.13).

Answer 1

The Lagrangian of a system consisting of two masses moving in a uniform gravitational field can be decomposed into two independent pieces: L = Lcm + Lrel. The Lcm term represents the motion of the center of mass of the system, while the Lrel term represents the relative motion between the masses.

Step-by-step explanation:

In the given question, we are asked to show how the Lagrangian of a system consisting of two masses moving in a uniform gravitational field can be decomposed.

In Equation (8.13), the Lagrangian is split into two independent pieces: L = Lcm + Lrel. The first term, Lcm, represents the motion of the center of mass of the system, which is not affected by the relative motion of the masses. The second term, Lrel, represents the relative motion between the masses.

In the case of two masses moving in a gravitational field and interacting via a potential energy U(r), the Lagrangian can be written as L = Lcm + Lrel, where Lcm = 0.5(M + m)V2 and Lrel = 0.5μv2 - U(r), with M and m being the masses of the two particles, V being the velocity of the center of mass, μ being the reduced mass of the system, v being the relative velocity between the masses, and r being the distance between them.