Exact invariants for a class of three-dimensional time-dependent classical Hamiltonians

An exact invariant is derived for three-dimensional Hamiltonian systems of $N$ particles confined within a general velocity-independent potential. The invariant is found to contain a time-dependent function $f_2(t)$, embodying a solution of a linear third-order differential equation whose coefficients depend on the explicitly known trajectories of the particle ensemble. Our result is applied to a one-dimensional time-dependent non-linear oscillator, and to a system of Coulomb interacting particles in a time-dependent quadratic external potential.