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The multidimensional energy surface of a cholesteric liquid crystal in aplanar cell is investigated as a function of spherical coordinates determiningthe director orientation. Minima on the energy surface correspond to the stablestates with particular director distribution. External electric and magneticfields deform the energy surface and positions of minima. It can lead to thetransitions between states, known as the Fr\'{e}edericksz effect. Transitionscan be continuous or discontinuous depending on parameters of the liquidcrystal which determine an energy surface. In a case of discontinuoustransition when a barrier between stable states is comparable with the thermalenergy, the activation transitions may occur and it leads to the modificationof characteristics of the Fr\'{e}edericksz effect with temperature withoutexplicit temperature dependencies of liquid crystal parameters. Minimum energypath between stable states on the energy surface for the Fr\'{e}edericksztransition is found using geodesic nudged elastic band method. Knowledge ofthis path, which has maximal statistical weight among all other paths, givesthe information about a barrier between stable states and configuration ofdirector orientation during the transition. It also allows one to estimate thestability of states with respect to the thermal fluctuations and their lifetimewhen the system is close to the Fr\'{e}edericksz transition.

Energy surface and minimum energy paths for Fr\'{e}edericksz transitions in bistable cholesteric liquid crystals