My understanding is that a system is a collection of particles. And according to Wikipedia, a closed system is one that does not allow transfer of matter in and out the system. However, my textbook says a closed system is one whose total mass is constant.
Are the two definitions equivalent? I always thought mass is just a measure of an object's resistance to being accelerated by a resultant force in an inertial frame of reference. How does change of mass relate to transfer of matter?
Furthermore, why do Wikipedia and my textbook say linear momentum is conserved in a closed and isolated system? I am able to prove linear momentum is conserved as long as the system is isolated, closure not required.
For any system at any time, $\vec p(t):=\sum_{i=1}^n\vec p_i(t)$ and $\vec p'(t)=\sum_{i=1}^n\vec p_i'(t)=\sum_{i=1}^n \vec F_i(t)$, where $\vec F_i(t)$ is the resultant force acting on the $i$th particle. $\sum_{i=1}^n\vec F_i(t)=\vec F(t)$, where $\vec F(t)$ is the resultant external force, because the internal forces exerted by particles among themselves cancel out by Newton's third law. Hence $\vec p'(t)=\vec F(t)$.
Also, $\vec p(t)=p_i(t)\vec i+p_j(t)\vec j+p_k(t)\vec k$ where $\vec i$,$\vec j$, and $\vec k$ are three linear independent vectors.
If the system is isolated, i.e $\vec F(t)= \vec0$, on $[t_1,t_2]$, then $\vec p'(t)=\vec 0$ and $p_i'(t)=p_j'(t)=p_k'(t)=0$ on $[t_1,t_2]$, which further implies $p_i(t)=C_i$, $p_j(t)=C_j$, and $p_k(t)=C_k$ on $[t_1,t_2]$. Hence $\vec p(t)=\vec C$ on $[t_1,t_2]$.
Am I wrong?
