  In this paper, we introduce two types of variational integrators, one
originating from the discrete Hamilton's principle while the other from
Galerkin variational approach. It turns out that these variational integrators
are equivalent to each other when they are used for integrating the classical
mechanical system with Lagrangian function
$L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible
symmetric constant matrix). They are symplectic, symmetric, possess
super-convergence order $2s$ (which depends on the degree of the approximation
polynomials), and can be related to continuous-stage partitioned Runge-Kutta
methods.
