Symmetries and Invariants of the Time-Dependent Oscillator Equation and the Envelope Equation

The single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant is fundamentally a result of the corresponding symmetry admitted by the oscillator equation with time-dependent frequency. A careful analysis of the admitted symmetries reveals a deeper connection between the non-linear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. The symmetries of the envelope equation enable a fast algorithm for finding matched solutions without using the conventional iterative shooting method.