The number of independent parameters required to define state of the system is called as degree of freedom. Take theory as some function which will provide output on the basis of some independent parameters.

Let us look at number of degrees of freedom in different theories of physics.

The natural world phenomena in day to day life can mostly be explained with just classical physics. In classical Mechanics we can just define state of system with two independent parameters per dimension(position and momenta/velocity).

Consider a system of N particles :

1. For one dimension: We need particles position and velocity/momenta to describe state of system. So number of degree of freedom is :2N ( N coordinates (of N particles) and (N momenta for N particles) ).
2. For two dimension: We need position (x,y) and velocity(\(v_{1},v_{2}\))/momenta ((\(p_{1},p_{2}\))), to describe state of system. So number of degree of freedom is :4N
3. For three dimension: We will require 3 positions(x,y,z) and 3 momenta/velocity : so number of degree of freedom is 6N.
4. For d-dimensions: We will require d positions (\(x_{1},x_{2},x_{3}....x_{d}\)) and 'd' momenta/velocity (\(p_{1},p_{2},p_{3}....p_{d}\)), so in total we have 2d.N number of degree of freedom.

The phenomena at very small scale can be explained with quantum mechanics. The state of system at that length scale can not be described by particle's position and momenta due to Heisenberg's uncertainty principle.