Physics sims retain data because some relationships and interactions between objects can span frames. You can of course add objects to an existing scene, but that would require you to renegotiate what is happening.

I have programmed a simple physics engine before, and I can say that it was very fun! I definitely recommend it as a learning experience. It will really improve your knowledge in applied math, and performance optimisation.

However: Writing a good physics engine is much harder than just doing collision detection. Simulating basic physics, such as a pool table. Is pretty straight forward. However, when processing interactions of more than 2 bodies (like a Jenga tower, or a solid ragdoll with hinges) : You should expect to run into alot of complex algebrical integration problems. One might say that this is what the likes of "PhysX" was designed to do. Using "physX" or "Havok" for a pool game would be overkill. 

So I would wholeheartedly recommend that you embark on writing your own Physics engine as a learning excersize:

0. Start by implementing some collision detection algorithms

1. Then make a 2D pool game.

2. Then make an angry birds clone. (If you've made it this far, you're a champ!)

3. Then try something simple in 3D (Maybe a racing game?)

1 hour ago, Amir_r_KB said:

O(n^2) might not be a big problem with my game

This really depends on the type of game you are making. If you use an spatial partioner (like an octree) , your interaction rate should be much lower. However, if O(n^2) is acceptable, then I think that you have no reason at all to optimise your calls to PhysX. Just use an existing engine, you probably don't have alot of physics to deal with...

1 hour ago, Amir_r_KB said:

Can you give an example?

Stacked objects which are considered "stable" will cease physics calculations, even though they are touching other (static) objects. Adding a new "dynamic" block on top of that can casue the entire stack to spontaneously "explode".