  Two-dimensional systems with time-dependent controls admit a quadratic
Hamiltonian modelling near potential minima. Independent, dynamical normal
modes facilitate inverse Hamiltonian engineering to control the system
dynamics, but some systems are not separable into independent modes by a point
transformation. For these "coupled systems" 2D invariants may still guide the
Hamiltonian design. The theory to perform the inversion and two application
examples are provided: (i) We control the deflection of wave packets in
transversally harmonic waveguides; and (ii) we design the state transfer from
one coupled oscillator to another.
