17.6: Normal Coordinates
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Landau writes Qα=ReCαeiωαt. (Actually he brings in an intermediate variable Θα, but we'll skip that.) These “normal coordinates” can have any amplitude and phase, but oscillate at a single frequency ¨Qα=−ω2αQα.
The components of the above vector equation read:
θ1=Q1/√3+Q2/√2+Q3/√6θ2=Q1/√3−2Q3/√6θ3=Q1/√3−Q2/√2+Q3/√6
It’s worth going through the exercise of writing the Lagrangian in terms of the normal coordinates:
recall the Lagrangian:
L=12˙θ21+12˙θ22+12˙θ23−12ω20θ21−12ω20θ22−12ω20θ23−12k(θ1−θ2)2−12k(θ3−θ2)2
Putting in the above expressions for the θα, after some algebra
L=12[˙Q21−ω20Q21]+12[˙Q22−(ω20+k)Q22]+12[˙Q23−(ω20+3k)Q23]
We’ve achieved a separation of variables. The Lagrangian is manifestly a sum of three simple harmonic oscillators, which can have independent amplitudes and phases. Incidentally, this directly leads to the action angle variables -- recall that for a simple harmonic oscillator the action I=E/ω, and the angle is that of rotation in phase space.