In this investigation, the formulation of trajectory coordinate constraints in terms of the absolute coordinates is developed for the use in the computer simulations of railroad vehicle system applications. Examples of trajectory coordinate constraints are the specified forward velocity of a vehicle or a wheelset along a curved track or specified yaw angle of a vehicle or a wheelset with respect to a body trajectory coordinate system. The relationship between the trajectory coordinates and the absolute coordinates is defined and then used to write the trajectory constraints in terms of the absolute coordinates at the position, velocity, and acceleration levels. The motion of the trajectory coordinate system can be uniquely defined in the track body coordinate system using the arc-length coordinate defined along the space curve. It is demonstrated that the use of the trajectory coordinates leads to simple linear constraint equations, while the trajectory constraint equations become highly nonlinear functions of the absolute coordinates. This is mainly due to the nature of the nonlinear relationship between the absolute Cartesian and trajectory coordinates. A numerical example is presented in order to demonstrate the use of the proposed formulations in the analysis of multibody railroad vehicle systems.