For the system in exercise 3, write down the equation of motion for (x_{1}^{2}+x_{2}^{2} equiv r^{2}) and
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For the system in exercise 3, write down the equation of motion for \(x_{1}^{2}+x_{2}^{2} \equiv r^{2}\) and then the corresponding Ehrenfest theorem equation and its transformed version under the continuous symmetries, in both the active and the passive sense.
Data From Exercise 3:-
Consider two harmonic oscillators of the same mass and frequency, with Hamiltonian
Find the continuous symmetries of the system and write down the resulting conserved charges as a function of the phase space variables. Then quantize the system, and show that the charges do indeed commute with the quantum Hamiltonian.
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