Two identical simple pendula are coupled together by a very weak force of attraction that...

Two identical simple pendula are coupled together by a very weak force of attraction that varies as the inverse square of the distance between the two particles. (This force might be the gravitational attraction between the two particles, for instance.) Show that, for small departures from the equilibrium configuration, the Lagrangian can be reduced to the same mathematical form, with appropriate constants, as that of the two identical coupled oscillators treated in Section 11.3 and in Problem (Hint: Consider Equation 11.3.9.)

Problem

In the system of two identical coupled oscillators shown in Figure 11.3.1, one oscillator is started with initial amplitude A0, whereas the other is at rest at its equilibrium position, so that the initial conditions are

Show that the amplitude of the symmetric component is equal to the amplitude of the antisymmetric component in this case and that the complete solution can be expressed as follows:

in which  and Δ= (ωb – ωa)/2. Thus, if the coupling is very weak so that K'« K, then ω will be very nearly equal to ωa = (K/m)1/2, and A is very small. Consequently, under die stated initial conditions, die first oscillator eventually comes to rest while the second oscillator oscillates with amplitude A0. Later, die system returns to die initial condition, and so on. Thus, the energy passes back and forth between die two oscillators indefinitely.