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It has been shown that the Cauchy problem for geodesics in the space ofK\"ahler metrics with a fixed cohomology class on a compact complex manifold$M$ can be effectively reduced to the problem of finding the flow of a relatedhamiltonian vector field $X_H$, followed by analytic continuation of the timeto complex time.  This opens the possibility of expressing the geodesic $\omega_t$ in terms ofGr\"obner Lie series of the form $\exp(\sqrt{-1} \, tX_H)(f)$, for localholomorphic functions $f$. The main goal of this paper is to use truncated Lieseries as a new way of constructing approximate solutions to the geodesicequation. For the case of an elliptic curve and $H$ a certain Morse functionsquared, we approximate the relevant Lie series by their first twelve terms,calculated with the help of Mathematica. This leads to approximate geodesicswhich hit the boundary of the space of K\"ahler metrics in finite geodesictime. For quantum mechanical applications, one is interested also on thenon-K\"ahler polarizations that one obtains by crossing the boundary of thespace of K\"ahler structures. Properties of the approximate geodesics and itsextensions are also studied using Mathematica.

A new approximation method for geodesics on the space of K\"ahler metrics using complexified symplectomorphisms and Gr\"obner Lie series