The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill[1] and David M. Fradkin,[2] is a conservation law used in the treatment of the isotropic multidimensional harmonic oscillator in classical mechanics.
The Fradkin tensor provides enough conserved quantities to make the oscillator's equations of motion maximally superintegrable.
[3] This implies that to determine the trajectory of the system, no differential equations need to be solved, only algebraic ones.
Similarly to the Laplace–Runge–Lenz vector in the Kepler problem, the Fradkin tensor arises from a hidden symmetry of the harmonic oscillator.
Suppose the Hamiltonian of a harmonic oscillator is given by with then the Fradkin tensor (up to an arbitrary normalisation) is defined as In particular,