We consider a Brownian particle in harmonic confinement of stiffness k, in
one dimension in the underdamped regime. The whole setup is immersed in a heat
bath at temperature T. The center of harmonic trap is dragged under any
arbitrary protocol. The thermodynamic uncertainty relations for both position
of the particle and current at time t are obtained using the second law of
thermodynamics as well as the positive semi-definite property of the
correlation matrix of work and degrees of freedom of the system for both
underdamped and overdamped cases.Comment: Minor revision, Accepted in EPJ
