# Degree of freedom and constraints

## 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

$s = 3$, $N = 1$ 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