What is the definition of momentum when a mass distribution $\rho(r,t)$ is given? This question is Edited after recieving comments.
What is the definition of momentum when a mass distribution $\rho(r,t)$ is given?
Assuming a particle as a point mass we know the definition of momentum as $p = mv$.
 I need a definition where it is assumed that point masses are not present.
 A: If you have multiple masses the total momentum is
$$ p = \sum_i m_i v_i $$
If you have a continuum distribution then you can proceed like follows.
You divide the continuum into small boxes where each part of the box has approximately same mass and velocity (this assumes some kind of smoothness of the distribution). Then you can obtain whole momentum with the above formula. Now letting the box sizes go to zero you obtain integral
$$ p(t) = \int \rho(r, t) v(r, t) dr $$
A: Counter-question:
What is the definition of mass, when a mass distribution given?
Strange question, isn't it?
Since you have the mass distribution you got to have the momentum distribution as well.
It is simply ρ*v. 
A: Motion of a non-point object can be described as a combination of translation (which only depends on the total mass of the object) and rotation (which depends on the mass distribution). Rotational momentum is defined as $\mathbf{L}= I \boldsymbol{\omega}$ where $I$ is a moment of inertia $I = \int_V \rho(\mathbf{r})\,d(\mathbf{r})^2 \, \mathrm{d}V\!(\mathbf{r})$
As you can see, the mass distribution defines the rotational momentum of a system.
