  The second-order differential equation for a damped harmonic oscillator can
be converted to two coupled first-order equations, with two two-by-two matrices
leading to the group $Sp(2)$. It is shown that this oscillator system contains
the essential features of Wigner's little groups dictating the internal
space-time symmetries of particles in the Lorentz-covariant world. The little
groups are the subgroups of the Lorentz group whose transformations leave the
four-momentum of a given particle invariant. It is shown that the damping modes
of the oscillator correspond to the little groups for massive and
imaginary-mass particles respectively. When the system makes the transition
from the oscillation to damping mode, it corresponds to the little group for
massless particles. Rotations around the momentum leave the four-momentum
invariant. This degree of freedom extends the $Sp(2)$ symmetry to that of
$SL(2,c)$ corresponding to the Lorentz group applicable to the four-dimensional
Minkowski space. The Poincar\'e sphere contains the $SL(2,c)$ symmetry. In
addition, it has a non-Lorentzian parameter allowing us to reduce the mass
continuously to zero. It is thus possible to construct the little group for
massless particles from that of the massive particle by reducing its mass to
zero. Spin-1/2 particles and spin-1 particles are discussed in detail.
