For the dynamics of a rigid body with a fixed point based on the quaternion and the corresponding generalized momenta, a displacement-based symplecti integration scheme for differential-algebraic equations is proposed and applied to the Lagrange's equations based on dependent generalized momenta. Numerical experiments show that the algorithm possesses such characters as high precision and preserving system invariants. More importantly, the generalized momenta based Lagrange's equations show unique advantages over the traditional Lagrange's equations in symplectic integrations.