A toy model for our expanding universe during the inflationary epoch consists of a circle of radius

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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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Modern Classical Mechanics

ISBN: 9781108834971

1st Edition

Authors: T. M. Helliwell, V. V. Sahakian

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