On the theory of time-dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type

Abstract

Writing the canonical variables (qT, pT) as (ωT), we develop a method for transforming the time-dependent Hamiltonian H=Aμν(t) ωμων+Bμων +C (t) to the time-independent form H̄= (1/2) δμνω̄μω̄ν using the linear transformation ω̄μ=sμ ν(t) ων +rμ(t). Differential equations are obtained for the parameters sμ ν and rμ. The transformed Hamiltonian enables the construction of an invariant I and an invariant matrix [Iμν]. These invariants apply to both the classical and quantum mechanical problems. The invariant I has the dynamical symmetry group SU(n), and this characterizes all systems with Hamiltonians of the form of H.