  The thermodynamic uncertainty relation provides a universal lower bound on
the product of entropy production and the fluctuations of any current. While
proven for Markov dynamics on a discrete set of states and for overdamped
Langevin dynamics, its status for underdamped dynamics is still open. We
consider a two-dimensional harmonically confined charged particle in a magnetic
field under the action of an external torque. We show analytically that,
depending on the sign of the magnetic field, the thermodynamic uncertainty
relation does not hold for the currents associated with work and heat. A strong
magnetic field can effectively localize the particle with concomitant bounded
fluctuations and low dissipation. Numerical results for a three-dimensional
variant and for further currents suggest that the existence of such a bound
depends crucially on the specific current.
