= Solution
Differentiate the <quadratic Lagrangian> with respect to coordinates and velocities. The equations are
$$
\boxed{\ddot x_1-2\Omega\dot x_2+(2k'/m-\Omega^2)x_1=0,\qquad
\ddot x_2+2\Omega\dot x_1+(2k/m-\Omega^2)x_2=0.}
$$
The velocity-cross terms are <Coriolis acceleration>[Coriolis accelerations]; moving $-\Omega^2x_i$ to the <force> side identifies the outward <centrifugal acceleration>. The remaining terms are elastic <Restoring force>[restoring forces]. The sign pair is fixed by the rotating-basis convention in part a.
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