space variables to . Having a system with a configuration or state space of: and a set of constraints where one or more constraints are in the Pfaffian form: does not guarantee the system is holonomic, as even though one differential has a coefficient of , there are still three degrees of freedom described in the configuration or state space. Transformation to independent generalized coordinates The holonomic constraint equations can help us easily remove some of the dependent variables in our system. For example, if we want to remove , which is a parameter in the constraint equation , we can rearrange the equation into the following form, assuming it can be done, and replace the in every equation of the system using the above function. This can always be done for general physical systems, provided that the derivative of is continuous, then by the implicit function theorem, the solution , is guaranteed in some open set. Thus, it is possible to remove all occurrences of the dependent variable . Suppose that a physical system has degrees of freedom. Now, holonomic constraints are imposed on the system. Then, the number of degrees of freedom is reduced to . We can use independent generalized coordinates () to completely describe the motion of the system. The transformation equation can be expressed as follows: Classification of physical systems In order to study classical physics rigorously and methodically, we need to classify systems. Based on previous discussion, we can classify physical systems into holonomic systems and non-holonomic systems. One of the conditions for the applicability of many theorems and equations is that the system must be a holonomic system. For example, if a physical system is a holonomic system and a monogenic system, then Hamilton's principle is the necessary and sufficient condition for the correctness of Lagrange's equation. See also Nonholonomic system Goryachev–Chaplygin top Pfaffian constraint Udwadia–Kalaba equation References Classical mechanics