Degree of freedom and constraints


This article is about Degree of freedom and constraints in classical mechanics. You can use this article for studying B.Sc, JAM , GATE and CSIR-NET Physics

Degree of freedom

Definition: – The minimum number of independent variables or co-ordinates required for specifying the position of a dynamical system consisting of one or more particle is called Degree of freedom.

For N number of  particles moving freely in s dimensional space degrees of freedom is represented by following equation.

f=Ns

Where N is the number of particles and s denote the dimensions of the particle.

If there are constraints then

f=Ns-k

where k is the number of constraints.

Examples:-

Consider a particle moving along x-direction in three dimensional space then in this case

s = 3N = 1 and k = 2

Constraint k = 2 means that particle can not move in y and z directions. So in this case degrees of freedom would be f=1.

Again degree of freedom would be f=1 when particle is moving in x-direction in one dimensional space.

Let us consider another case where two particles are moving freely in xy plane. Here in this case





\begin{gathered} s = 2 \hfill \\ N = 2 \hfill \\ k = 0 \hfill \\ \end{gathered}

here k=0 because particles are not constrained to move in xy-plane.

Now degrees of freedom is defined as f=Ns-k which gives us f=4

Constraints

  • A constrained motion is a motion which can not proceed arbitrary in any manner.
  • Particle motion can be restricted to occur (1) along some specified path (2) on surface (plane or curved) arbitrarily oriented in space.
  • Imposing constraints on a mechanical system is done to simplify the mathematical description of the system.
  • Constraints expressed in the form of equation f(x1,y1,z1,......,xn,yn,zn :t)=0 are called  holonomic constraints.
  • Constraints not expressed in this fashion are called non-holonomic constraints.
  • Scleronomic constraints are independent of time.
  • Constraints containing  time explicitely are called  rehonomic.

Therefore a constraint is either

          Scleronomic where constraints relations does not depend on time or rheonomic where constraints relations depends explicitly on time

or

           Holonomic where constraints relations can be made independent of velocity or non-holonomic where these relations are irreducible functions of velocity

Constraints types of some physical systems are given below in the table

constraints

More about constraints

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
n=3N-k                                  (1)
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
f(r1,r2,……t)=0                          (2)
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 ri and rj are the position vectors of the i’th and j’th particles then , the distance between the is given by
|ri-rj|=cij                                   (3)
The constraints which are not expressible in the form of equation 2 are called non-holonoic for example, the motion of a particle placed on the surface of a sphere of radius a will be described as
|r|≥a or, r-a≥0