CONSTRAINT RELATIONS

In classical mechanics, a constraint is a relation between coordinates and momenta (and possibly higher derivatives of the coordinates). In other words, a constraint is a restriction on the freedom of movement of a system of particles.

The only difference between a constraint equation and eg a conservation equation is that a conservation equation is physics, but a constraint equation is geometry.

With example, a system of three pulleys at heights p q and r, we get physics equations (usually F = ma) for each pulley, but the "a" in F = ma is different for each pulley (in fact, it's p'' q'' and r'' respectively)
so we need a geometric equation relating p q and r …
usually this simply tells us the length of the string in terms of p q and r …
since we know that that length is constant, we can differentiate once (or twice) to get a neat "constraint equation"

The following are different examples of Constraint Relations. 

Two masses tied to a string going over a friction less pulley m1 > m2 as shown in Fig. 3.1.

If 'a' is the common acceleration of the masses and 'T' the tension in the string, then, for mass m1

                  

Similarly for mass m2, the net force which gives 

          

From Eqs. (i) and (ii) we get,

                                                        

                                               and 

Two masses in contact :

 Figure 3.2 shows two masses m1 and m2 in contact placed on a horizontal frictionless surface. A force F is applied as shown and as a result the masses move with acceleration, which is given by

                              

The force on m1 is F and the contact force on m2 is 

                                                             

                                

Three masses in contact:

 If three masses m1, m2 and m3 are placed in contact and force F is applied to mass m1,  the three masses move with acceleration (see Fig. 3.3)

            

                         

The force acting on mass m1 is F. The contact forces acting of masses m2 and m3 respectively are

                                

                          

                        

                                                               and