Normal Modes Solution for Two-Body Oscillator

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SUMMARY

The discussion focuses on the normal modes solution for a coupled two-body oscillator, specifically the equations for the displacements x1(t) and x2(t). The general solution is expressed with four adjustable constants: C1-, C1+, ψ-, and ψ+. The inquiry centers around the relationship between the phase constants ψ1- and ψ2-, questioning why ψ2- cannot be expressed as a function of ψ1- and the constants C1- and C1+. This highlights the constraints in maintaining a "pure" in-phase normal mode solution.

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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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