  We show how the recently developed theory of geodesic transport barriers for
fluid flows can be used to uncover key invariant manifolds in externally
forced, one-degree-of-freedom mechanical systems. Specifically, invariant sets
in such systems turn out to be shadowed by least-stretching geodesics of the
Cauchy-Green strain tensor computed from the flow map of the forced mechanical
system. This approach enables the finite-time visualization of generalized
stable and unstable manifolds, attractors and generalized KAM curves under
arbitrary forcing, when Poincare maps are not available. We illustrate these
results by detailed visualizations of the key finite-time invariant sets of
conservatively and dissipatively forced Duffing oscillators.
