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!