  Despite intense interest in realizing topological phases across a variety of
electronic, photonic and mechanical platforms, the detailed microscopic origin
of topological behavior often remains elusive. To bridge this conceptual gap,
we show how hallmarks of topological modes - boundary localization and
chirality - emerge from Newton's laws in mechanical topological systems. We
first construct a gyroscopic lattice with analytically solvable edge modes, and
show how the Lorentz and spring restoring forces conspire to support very
robust "dangling bond" boundary modes. The chirality and locality of these
modes intuitively emerges from microscopic balancing of restoring forces and
cyclotron tendencies. Next, we introduce the highlight of this work, a very
experimentally realistic mechanical non-equilibrium (Floquet) Chern lattice
driven by AC electromagnets. Through appropriate synchronization of the AC
driving protocol, the Floquet lattice is "pushed around" by a rotating
potential analogous to an object washed ashore by water waves. Besides hosting
"dangling bond" chiral modes analogous to the gyroscopic boundary modes, our
Floquet Chern lattice also supports peculiar half-period chiral modes with no
static analog. With key parameters controlled electronically, our setup has the
advantage of being dynamically tunable for applications involving arbitrary
Floquet modulations. The physical intuition gleaned from our two prototypical
topological systems are applicable not just to arbitrarily complicated
mechanical systems, but also photonic and electrical topological setups.
