A deeper understanding of nonequilibrium phenomena is needed to reveal the
principles governing natural and synthetic molecular machines. Recent work has
shown that when a thermodynamic system is driven from equilibrium then, in the
linear response regime, the space of controllable parameters has a Riemannian
geometry induced by a generalized friction tensor. We exploit this geometric
insight to construct closed-form expressions for minimal-dissipation protocols
for a particle diffusing in a one dimensional harmonic potential, where the
spring constant, inverse temperature, and trap location are adjusted
simultaneously. These optimal protocols are geodesics on the Riemannian
manifold, and reveal that this simple model has a surprisingly rich geometry.
We test these optimal protocols via a numerical implementation of the
Fokker-Planck equation and demonstrate that the friction tensor arises
naturally from a first order expansion in temporal derivatives of the control
parameters, without appealing directly to linear response theory