Normal Modes Solution for Two-Body Oscillator

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In the discussion on the normal modes solution for a coupled two-body oscillator, the general solution is presented with equations for x1(t) and x2(t), incorporating constants C1-, C1+, ψ-, and ψ+. The inquiry focuses on why ψ2- cannot be expressed as a function of ψ1- and other constants to maintain a "pure," in-phase normal mode solution. The question suggests a need for clarity on the relationship between these variables in the context of normal modes. Participants are encouraged to provide insights or explanations regarding this aspect of coupled oscillators. The discussion highlights the complexity of determining the relationships between oscillation parameters in two-body systems.
Migdal
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Hello!

For a coupled two-body oscillator we write the general solution as:
x1(t)=C1-Cos[ω-t+ψ1-]+C1+Cos[ω+t+ψ1+]
x2(t)=C2-Cos[ω-t+ψ2-]+C2+Cos[ω+t+ψ2+]
Where we determine C1-/C2- and C1+/C2+ from the normal mode condition.

We call ψ1-=ψ2-=ψ- and ψ1+= ψ2+=ψ+, and we end up with 4 adjustable constants: C1-,C1+, ψ-, ψ+.

Why is that? Why can't ψ2- be a function of ψ1-,( ψ1+ maybe), C1- and C1+, such that ψ2-(C1+=0)=ψ1-, in order to keep the "pure", in phase, normal mode solution? The same for ψ2+.

Thank you in advance!
 
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Can anybody give me any kind of answer? Maybe it's an inappropriate question.
 

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