instead of only a configuration space, is called rheonomic. Many systems can be described either scleronomically or rheonomically. For example, the total allowable motion of a pendulum can be described with a scleronomic constraint, but the motion over time of a pendulum must be described with a rheonomic constraint. The state space is the configuration space, plus terms describing the velocity of each term in the configuration space. The state-time space adds time . Examples Gantry crane As shown on the right, a gantry crane is an overhead crane that is able to move its hook in 3 axes as indicated by the arrows. Intuitively, we can deduce that the crane should be a holonomic system as, for a given movement of its components, it doesn't matter what order or velocity the components move: as long as the total displacement of each component from a given starting condition is the same, all parts and the system as a whole will end up in the same state. Mathematically we can prove this as such: We can define the configuration space of the system as: We can say that the deflection of each component of the crane from its "zero" position are , , and , for the blue, green, and orange components, respectively. The orientation and placement of the coordinate system does not matter in whether a system is holonomic, but in this example the components happen to move parallel to its axes. If the origin of the coordinate system is at the back-bottom-left of the crane, then we can write the position constraint equation as: Where is the height of the crane. Optionally, we may simplify to the standard form where all constants are placed after the variables: Because we have derived a constraint equation in holonomic form (specifically, our constraint equation has the form where ), we can see that this system must be holonomic. Pendulum As shown on the right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. This system is holonomic because it obeys the holonomic constraint where is the position of the weight and is length of the string. Rigid body The particles of a rigid body obey the holonomic constraint where , are