# Is linear momentum of an open system conserved?

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?

• One could make the statement of a closed system stronger by stipulating that no mass-energy enters or leaves the system. Tracking all the particles only suffices if the particles don't interact with their environment. A sealed piston expanding under load is not a closed system: mechanical work on the load crosses the boundary, so to close the system we must include the load. A sealed flask of water immersed in a heat source is not a closed system: heat from the source crosses the boundary, so to close the system we must include the source. Etc.
– g s
Commented Mar 27, 2022 at 5:04
• @gs Some sources use "isolated system" for what you are describing. Under this definition, a sealed flask of water immersed in a heat source is a closed system but not an isolated system.
– Chris
Commented Mar 27, 2022 at 21:17

The two definitions are not equivalent- the former is strictly stronger than the latter. Although you will get a lot of the same results mathematically using either definition.

If the system is open, the momentum of the system is trivially not conserved if any entering/exiting particles have momentum of their own. Your proof fails in assuming that the sums $$\sum_{i=1}^{n}$$ are over a fixed number of particles. This is true for a closed system but not for an open system.

• I thought a system is a collection of particles. And a system with a different collection of particles is another system. How can some particle enter a system? Isn't a system just a collection of particles? The observer can group things together mentally.
– TFR
Commented Mar 27, 2022 at 4:55
• @TFR A system in general is not necessarily a consistent collection of particles. Consider if one was interested in the airflow over a wing: all the particles in the system are continuously leaving and being replaced! If one tracks all the constituent parts and doesn't allow them to interact with anything that isn't one of the tracked parts, one has a closed system. If one allows tracked parts either to leave or to interact with non-tracked parts, and/or if one tracks new parts, one has an open system.
– g s
Commented Mar 27, 2022 at 5:29
• @gs So a better definition for a system is a mapping between time to a set of particles that satisfy a fixed property? And a closed system is one whose set of particles is constant with respect to time and that there is no transfer of energy to or from it?
– TFR
Commented Mar 27, 2022 at 17:42
• @TFR I'm not sure what you mean by the first sentence. The second sentence is correct. I would use "parts" not "particles" to reflect that it's not always useful to model anything that would be reasonably described as a particle, e.g. if your system is the sun and planets, you wouldn't try to model all the molecules. I would replace "no transfer" with "negligible transfer" to reflect that some environmental influence is unavoidable in real life but can sometimes be ignored in the model (e.g. if your system is the sun and planets, you wouldn't include distant stars in your model).
– g s
Commented Mar 27, 2022 at 18:04
• A system is any set of processes and objects under consideration. It could be "these particular particles" or "the particles in my box" or "the box and its contents" or "fox and rabbit populations in South Dakota" or "the local galactic supergroup".
– g s
Commented Mar 27, 2022 at 18:11