Solution To Suggested Advanced Lagrangian Problem

 For each of the frequency solutions applied to the linear tri-atomic molecule, show the relation between the normalized amplitudes: A1, A2 and A3

Solution:

For:  w1  =  0,  A1  =  A2  =  A3   (Pure translation)

For:   w2  = Ö(k/ m),   A1  =  - A3,   A2  = 0 

For:   w3  =  Ö{ k/ m (1 + 2m/ M)},   A1  = A3   = -M A2/ 2m