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For a coupled two-body oscillator we write the general solution as:

x1(t)=C1_{-}Cos[ω_{-}t+ψ_{1-}]+C1_{+}Cos[ω_{+}t+ψ_{1+}]

x2(t)=C2_{-}Cos[ω_{-}t+ψ_{2-}]+C2_{+}Cos[ω_{+}t+ψ_{2+}]

Where we determine C1_{-}/C2_{-}and C1_{+}/C2_{+}from the normal mode condition.

We call ψ_{1-}=ψ_{2-}=ψ_{-}and ψ_{1+}= ψ_{2+}=ψ_{+}, and we end up with 4 adjustable constants: C1_{-},C1_{+}, ψ_{-}, ψ_{+}.

Why is that? Why can't ψ_{2-}be a function of ψ_{1-},( ψ_{1+}maybe), C1_{-}and C1_{+}, such that ψ_{2-}(C1_{+}=0)=ψ_{1-}, in order to keep the "pure", in phase, normal mode solution? The same for ψ_{2+}.

Thank you in advance!

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# Normal modes solution

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