This paper is concerned with several possible formulations of constrained Hamiltonian systems and their numerical solutions by using low order symplectic and nonsymplectic methods. The systems under consideration are separable systems with a Hamiltonian of type \({H(p,q) = (1/2) p^ Tp + F(q)}\) with smooth holonomic constraints given by \(g(q) = 0\).
In section 2 the existence of a (local) free parametrization of the variety \(g(q) = 0\) is shown, that allows to describe the system in Hamiltonian form in terms of the free parameters. Here the treatment is similar to the one used to describe conservative systems with Lagrangian coordinates in the context of Analytical Dynamics.
In section 3 Hamiltonian formulations of the problem are presented in the general \((p,q)\) variables with linear and nonlinear constraints by using Dirac’s theory of constrained Hamiltonian systems.
Some numerical results obtained with different formulations of the pendulum problem and symplectic and non-symplectic methods are presented. It is shown that, in general, the conservation of energy in the long term integration by means of symplectic methods is generally better than its corresponding to the non-symplectic methods. However the authors show that a weakly Hamiltonian formulation provides better results even with nonsymplectic methods.