A toy model for our expanding universe during the inflationary epoch consists of a circle of radius \(r(t)=r_{0} e^{\omega t}\) where we are confined on the one-dimensional world that is the circle. To probe the physics, imagine two point masses of identical mass \(m\) free to move on this circle without friction, connected by a spring of force-constant \(k\) and relaxed length zero, as depicted in the figure.

(a) Write the Lagrangian for the two-particle system in terms of the common radial coordinate \(r\), and the two polar coordinates \(\theta_{1}\) and \(\theta_{2}\). Do not implement the radial constraint \(r(t)=r_{0} e^{\omega t}\) yet.

(b) Using a Lagrange multiplier for the radial constraint, write four equations describing the dynamics. In this process, show that

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alr = 1 a = -wr. (c) Consider the coordinate relabeling " 3=01-02. Show that the equations of motion of part (b) for the two angle variables 0 and 0 can be rewritten in a decoupled form as = ; B = CB+C3B; where C1, C2 and C3 are constants that you will need to find. (d) identify a symmetry transformation {St, Sa, 88} for this system. Find the associated conserved quantity. What would you call this conserved quantity? (e) Then find a(t) and 3(t) using equations 6.186 and 6.187. Use the boundary conditions a(0)=ao, (0) = 0, 3(0) = 0, 8(0) = C. What is the effect of the expansion on the dynamics? Note: This conclusion is the same as in the more realistic three-dimensional cosmological scenario.