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Physics LibreTexts

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=Q1/32Q3/6θ3=Q1/3Q2/2+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˙θ2312ω20θ2112ω20θ2212ω20θ2312k(θ1θ2)212k(θ3θ2)2

Putting in the above expressions for the θα, after some algebra

L=12[˙Q21ω20Q21]+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.


This page titled 17.6: Normal Coordinates is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.

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