Consider a particle of mass \(m\) moving in three dimensions but constrained to the surface of the paraboloid \(z=\alpha\left((x-1)^{2}+(y-1)^{2}\right)\). The particle is also subject to the spring potential \(U(x, y, z)=(1 / 2) k\left(x^{2}+y^{2}\right)\).

(a) Show that the Lagrangian of the system is given by

\[L=\frac{1}{2} m \dot{x}^{2}\left(1+4 \alpha^{2}\right)+\frac{1}{2} m \dot{y}^{2}\left(1+4 \alpha^{2}\right)+\frac{1}{2} 8 m \alpha^{2} \dot{x} \dot{y}-\frac{1}{2} k\left(x^{2}+y^{2}\right)\]

to quadratic order in \(x, y, \dot{x}\), and \(\dot{y}\) - assuming the displacement from the origin is small.

(b) Find the normal modes, eigenfrequencies and eigenvectors.