  We take a new look at the details of symplectic motion in solenoid and
bending magnets and rederive known (but not always well-known) facts. We start
with a comparison of the general Lagrangian and Hamiltonian formalism of the
harmonic oscillator and analyze the relation between the canonical momenta and
the velocities (i.e. the first derivatives of the canonical coordinates). We
show that the seemingly non-symplectic transfer maps at entrance and exit of
solenoid magnets can be re-interpreted as transformations between the canonical
and the mechanical momentum, which differ by the vector potential.
  In a second step we rederive the transfer matrix for charged particle motion
in bending magnets from the Lorentz force equation in cartesic coordinates. We
rediscover the geometrical and physical meaning of the local curvilinear
coordinate system. We show that analog to the case of solenoids - also the
transfer matrix of bending magnets can be interpreted as a symplectic product
of 3 non-symplectic matrices, where the entrance and exit matrices are
transformations between local cartesic and curvilinear coordinate systems.
  We show that these matrices are required to compare the second moment
matrices of distributions obtained by numerical tracking in cartesic
coordinates with those that are derived by the transfer a matrix method.
