Symmetry groups and conserved quantities for the harmonic oscillator

Abstract

The complete eight-parameter symmetry group of the one-dimensional harmonic oscillator is investigated using the fact that the system is describable by a variational principle. It is found that only a five-parameter subgroup leaves the action integral invariant, thus yielding five conserved quantities, only two of which are functionally independent. These two conserved quantities determine the solutions, and correspond to a two-parameter Abelian subgroup. The author also shows that if a conserved quantity corresponds to a symmetry group by Noether's theorem, then the group transforms any solution into another solution possessing the same value of the conserved quantity. In addition, we find how the Lagrangian transforms under symmetry groups which do not preserve the action integral, leading to certain alternative Lagrangians for the harmonic oscillator.