Question

Why phase space is required in classical mechanics and statistical mechanics?, as this is not a real space. what probelm we would suffer if the phase space is not assumed or cosidered?

Answer 1

In these cases the independent variables are generalised co-ordinates along with time using which we find that velocity is a dependent variables and hence not of much use in terms of representation. Rather mechanical state of a system cann be descirbed completely in proper manner by using momentum and position co-ordinates.

Now, after denoting momentum as co-ordinate in case of representation of systems, the configuration space is no longer adequate to represent the history of the system. The path adopted by the system during it's motion must now be represented by a space of 6N dimensions instead of 3 dimentional configuration space. Where each particle contributes one dimension for each position co-ordinate and one for each momentum component. This new space is termed as Phase Space and the reason for it's requirement is stated above in the explanation.

If this is not considered we can't formulate the Hamiltonian description of a system also there can't be any canocial equations of motion making classical mechanics obsolete. Same applies to Statistical Mechanics as without this assummed space we can't define ensemebles which form the basis of statistical mechanics.