Indirect Analytic Representation of Foucault's Pendulum

A complex representation of the equations of motion of the Foucault's pendulum is considered and the inverse problem is solved to derive an indirect analytic representation. Both real L R (i) and imaginary L I (i) parts of the derived complex valued Lagrangian are found to reproduce the equations of motion via the Euler-Lagrange equations. The expressions for L R (i) and L I (i) are not connected by a gauge term thereby forming a set of inequivalent Lagrangians. In an appropriate limit L I (i) is found to reproduce the Lagrangian obtained by implementing the usual Coriolis theorem while in some other limit L R (i) and L I (i) give the indirect and the direct analytic representations for a set of two uncoupled harmonic oscillators.