 Solution

Use rotating basis vectors with ${\dot{\mathbf{e}}}_1=\Omega {\mathbf{e}}_2$, ${\dot{\mathbf{e}}}_2=-\Omega {\mathbf{e}}_1$. The inertial velocity is $({\dot{x}}_1-\Omega x_2){\mathbf{e}}_1+({\dot{x}}_2+\Omega x_1){\mathbf{e}}_2$. The two fixed rotating anchors are $(0,\pm a)$, so the spring lengths are ${\ell }_{\pm }=\sqrt{x_1^2+(x_2\mp a{)}^2}$. Combining kinetic and elastic energies givesA constant elastic offset has no dynamical effect.