Coupled equation transformation

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mikengan
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can anyone help explain the mechanism of coupled equation transformation and when and how can i transform similar equations?
 
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  • Handwritten-style typed derivation: Let ω' = ω sin λ. Coupled equations ẍ = −(g/L)x + 2ω'ẏ and ÿ = −(g/L)y − 2ω'ẋ, with a highlighted rotation transformation x = x' cos ω't + y' sin ω't, y = −x' sin ω't + y' cos ω't, and derivative expressions for ẋ and ẏ.
    couple equation.png
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