Newton's laws in non inertial frame Suppose we take a two particle system in CM frame such that the particles are connected by a massless spring and a force $F$ is applied on one particle ($m_1$ and $m_2$. Let $F$ acting on $m_2$). Now if we try to study application of Newton's laws in accelerated CM frame we use pseudo force to use second law? Does spring force become external or internal in free body diagrams of individual particles?
As per me, I would prefer thinking spring force acting on individual particles even in CM frame. Only thing a spring force won't do is to accelerate CM. I need your inputs to know if my understanding of non inertial frames and Newton's laws application is alright? We may take some other examples of some em variable force acting between two particles in place of a spring. The confusion persists!
 A: Let me recall what Newton's laws are, to start with:
1) In the universe exists at least one reference frame (that we call inertial) where $\textbf{v}= \textrm{const.}$ whenever no external interactions act on the particle. All other reference frames (if any) moving at constant speed wrt this very one will be inertial as well.
2) In the above reference frames (if any) whenever external interactions $\textbf{F}(\textbf{x},\dot{\textbf{x}})$ act, it holds $\textbf{F}(\textbf{x},\dot{\textbf{x}}) = \dot{\textbf{p}}$.
3) The sum of all mutual interactions acting on a system of particles is zero.
If you have non-inertial frames the best way to calculate what the dynamics are is to start from the positions of the particles and take derivatives, to make velocities and accelerations appear. Once you take such derivatives (twice) you will see that some of them involve the reference frames moving (which we call non-inertial forces) and some others are exactly nothing but $\dot{\textbf{p}}$. Therefore you will end up with an equation like:
$$
\dot{\textbf{p}} = \sum (\textrm{non-inertial forces}) +  \textbf{F}(\textbf{x},\dot{\textbf{x}})
$$
which in principle splits the real forces from the non-inertial ones. Then what you may want to call which is up to you; nevertheless the dynamics will be given by the solutions of the above.
