  Accelerating controlled thermodynamic processes requires an auxiliary
Hamiltonian to steer the system into instantaneous equilibrium states. An extra
energy cost is inevitably needed in such finite-time operation. We recently
develop a geodesic approach to minimize such energy cost for the shortcut to
isothermal process. The auxiliary control typically contains momentum-dependent
terms, which are hard to be experimentally implemented due to the requirement
of constantly monitoring the speed. In this work, we employ a variational
auxiliary control without the momentum-dependent force to approximate the exact
control. Following the geometric approach, we obtain the optimal control
protocol with variational minimum energy cost. We demonstrate the construction
of such protocol via an example of Brownian motion with a controllable harmonic
potential.
