Mass-weighted oscillator symmetries
= Mass-weighted oscillator symmetries
For independent coordinates with $L=\sum_i m_i(\dot x_i^2-\omega^2x_i^2)/2$, the variables $X_i=\sqrt{m_i}x_i$ give an isotropic oscillator even at unequal masses. Rotations of the $X_i$ are continuous symmetries. Their conserved <Noether theorem>[Noether charges] $X_iP_j-X_jP_i$ therefore predate any equality of physical masses. Equal masses alone cannot create new invariances of arbitrary additional interactions.