 Solution

The kinetic energy is $m({\dot{\eta }}_1^2+{\dot{\eta }}_2^2)/2$. The Euler–Lagrange equations giveFor a normal mode $\eta =A{e}^{i\omega t}$, the stiffness matrix isSolving this quadratic yieldsBoth are positive for positive spring constants. A corresponding amplitude ratio is $A_2/A_1=(j+k-m{\omega }_{\pm }^2)/k$; the lower-frequency mode has the masses moving in the same direction and the higher-frequency mode in opposite directions.