QUE/CM-07009] A small oscillation problem in two dimensions

 Consider a particle of mass \(m\) moving in two dimensions in a potential \[ V(x,y) =   
     \frac{k}{2}(5 x^2 + 2 x y + 5 y^2)\]

1.   At what point \((x_0,y_0)\) is the particle in stable  equilibrium?
2.   Give the Lagrangian appropriate for small oscillations about this equilibrium   position.
3.   Find the normal frequencies of vibration in (b).
4. Write the Lagrangian in terms of normal coordinates and verify that it takes the form \[\mathscr{L} = \frac{1}{2}(\dot{Q}_1^2 + \dot{Q}_2^2) -\frac{1}{2}( Q_1^2 + Q_2^2)\]