The number of independent coordinates required to specify the configuration 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 specify the configuration of a system with one independent coordinate (position) per dimension for each particle.
Consider a system of N particles :
- For one dimension: We need each particle's position (x) to specify the configuration of the system. So number of degree of freedom is :N ( N coordinates of N particles ).
- For two dimension: We need position (x,y) of each particle to specify the configuration of the system. So number of degree of freedom is :2N
- For three dimension: We will require 3 positions(x,y,z) for each particle : so number of degree of freedom is 3N.
- For d-dimensions: We will require d positions (\(x_{1},x_{2},x_{3}....x_{d}\)) for each particle, so in total we have d.N number of degree of freedom.
Note: Momenta/velocities are not counted as separate degrees of freedom. They are needed, along with the positions, to specify the full dynamical state of the system (a point in phase space), so the state of N particles in d dimensions is given by 2d.N numbers, i.e. twice the number of degrees 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.