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 axe found to reproduce the equations of motion via the Euler-Lagrange equations. The expressions for L-R((i)) and L-I((i)) axe not connected by a gauge term thereby forming a set of inequivalent Lagrangians. In an appropriate limit L-R((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.