  Quaternions were appeared through Lagrangian formulation of mechanics in
Symplectic vector space. Its general form was obtained from the Clifford
algebra, and Frobenius' theorem, which says that "the only finite-dimensional
real division algebra are the real field ${\bf R}$, the complex field ${\bf C}$
and the algebra ${\bf H}$ of quaternions" was derived. They appear also through
Hamilton formulation of mechanics, as elements of rotation groups in the
symplectic vector spaces. Quaternions were used in the solution of
4-dimensional Dirac equation in QED, and also in solutions of Yang-Mills
equation in QCD as elements of noncommutative geometry. We present how
quaternions are formulated in Clifford Algebra, how it is used in explaining
rotation group in symplectic vector space and parallel transformation in
holonomic dynamics. When a dynamical system has hysteresis, pre-symplectic
manifolds and nonholonomic dynamics appear. Quaternions represent rotation of
3-dimensional sphere ${\bf S}^3$. Artin's generalized quaternions and
Rohlin-Pontryagin's embedding of quaternions on 4-dimensional manifolds, and
Kodaira's embedding of quaternions on ${\bf S}^1\times {\bf S}^3$ manifolds are
also discussed.
