A mechanical system is observed that is acted upon by potential forces in the presence of bonds $f(q)=0$. At the same time, impulses are introduced by relations (1) in which $\lambda$ are undetermined multipliers. The theorem is proved: the functions $p(t)$ and $q(t)$, which determine the motions of the system with n$-1$ degrees of freedom, satisfy Hamilton's equations. This observation was applied to the determination of the motion of a rigid body with fixed points. Within the quaternion space, the space of excess coordinates is considered. The Hamiltonian function is determined. For the case when the force field is invariant with respect to a group of rotations around some fixed axis, the cosines that determine the orientation are expressed by relations (2). The system of Hamilton's differential equations that has a linear integral $F=\Sigma f(q)p$ lowers the order by one, which is applied to the case of rotational motion of a rigid body for which equations (3) are reduced to the form (5).