AMBIGUITIES IN THE ASSOCIATION BETWEEN SYMMETRIES AND CONSERVATION LAWS IN THE PRESENCE OF ALTERNATIVE LAGRANGIAN REPRESENTATIONS

We identify two alternative Lagrangian representations for the damped harmonic oscillator characterised by a frictional coefficient gamma. The first one is explicitly time independent while the second one involves time parameter explicitly. With separate attention to both Lagrangians we make use of the Noether theorem to compute the variational symmetries and conservation laws in order to study how association between them changes as one goes from one representation to the other. In the case of time-independent representation squeezing symmetry leads to conservation of angular momentum for gamma = 0, while for the time-dependent Lagrangian the same conserved quantity results from rotational invariance. The Lie algebra (g) of the symmetry vectors that leaves the action corresponding to the time-independent Lagrangian invariant is semi-simple. On the other hand, g is only a simple Lie algebra for the action characterised by the time-dependent Lagrangian.