Sometimes motion of a particle or system of particles is restricted by one or more conditions. The limitations on the motion of the system are called constraints. The number of co-ordinates needed to specify the dynamical system becomes smaller when constraints are present in the system. Hence the degree of freedom of a dynamical system is defined as the minimum number of independent co-ordinates required to simplify the system completely along with the constraints. Thus if k is the number of constraints and N is the number of particles in the system possessing motion in three dimensions then the number of degrees of freedom are given by
thus the above system has n degrees of freedom.
Constraints may be classified in many ways. If the condition of constraints can be expressed as equations connecting the co-ordinates of the particles and possibly the time having the form
then constraints are said to be holonoic and the simplest example of holonomic constraints is rigid body. In case of rigid body motion the distance between any two particles of the body remains fixed and do not change with the tie. If and are the position vectors of the i’th and j’th particles then , the distance between the is given by
The constraints which are not expressible in the form of equation 2 are called non-holonomic for example, the motion of a particle placed on the surface of a sphere of radius a will be described as
Constraints can further be described as (i) rehonoic and(ii) scleronoous. In rehonomous constraints the equation of constraints contains time as explicit variable while in case of scleronomous constraints they are not explicitly dependent on time.