  In this paper, we systematically construct two classes of
structure-preserving schemes with arbitrary order of accuracy for canonical
Hamiltonian systems. The one class is the symplectic scheme, which contains two
new families of parameterized symplectic schemes that are derived by basing on
the generating function method and the symmetric composition method,
respectively. Each member in these schemes is symplectic for any fixed
parameter. A more general form of generating functions is introduced, which
generalizes the three classical generating functions that are widely used to
construct symplectic algorithms. The other class is a novel family of energy
and quadratic invariants preserving schemes, which is devised by adjusting the
parameter in parameterized symplectic schemes to guarantee energy conservation
at each time step. The existence of the solutions of these schemes is verified.
Numerical experiments demonstrate the theoretical analysis and conservation of
the proposed schemes.
