Why Newton's equation of motion is time reversally invariant (TRI)? I am really perplexed by the fact that Newton's equation is time reversal?
Newton's equation of motion is time reversally invariant, evident from the equation itself:
$$m\dfrac{d^2x}{dt^2} = F(x).$$
My question is why?
Is there some deep reason they come out to be time reversally invariant(may be connection to Principle of Least action, which is a global picture by the way, instead of being local which is the case of Newton's equation)? or connection to something else(which is more evident)?    
In the equation of motion(eom) because of the acceleration term, instead of velocity or other(which defy TRI). Is there a reason of it coming out to be like this? Links from geometry(variational principle) or where I see clearly that, it has to be like this(very basic and physically intuitive).     
A bit detailed explanation will be of great use(origin of such symmetry here).
 Forgive me, if question is unclear(make it clear, if asked) or if it has been asked(I checked but not my question), any help is highly appreciable.
 A: Newton's equation of motion, as it was phrased in his time and even as it is taught today, is not necessarily invariant under time reversal for it is 
\begin{equation}
m \vec{a} = \vec{F}
\end{equation}
In Newton's mind (from what I have read at least) and in general for this equation to be of maximum use, $\vec{F}$ can be anything as long as it can be extracted from the combination of trajectory data and the above equation. Hence, it can depend explicitly on time and, in principle, depend on any order time derivatives of the position, like velocity or acceleration or the jerk (time derivative of acceleration).
Examples in physics and engineering are taught in most undergraduate courses:


*

*Fluid friction force is in general a non trivial function of velocity (in best scenario has the form $\vec{F} = -\gamma \dot{\vec{r}}$).

*The magnetic force experienced by a charged particle by an imposed magnetic field is of the form $\vec{F} = q \dot{\vec{r}} \times \vec{B} $.

*The Abraham-Lorentz force which attempts to account for the action of radiation of a particle on its own motion has a term in the force that goes as $\vec{F} \propto \dddot{\vec{r}}$.
In all the above examples Newton's equation of motion is not invariant under time reversal (one could mention that for the Lorentz force, the equation of motion is invariant under CT transformation i.e. time reversal and charge conjugation).
So what do people mean when they say that "Newton's equations of motion are reversible"? Well, they only refer to a small set of forces, so-called conservative forces, which only depend on the (relative) positions of the particles.
There is strong evidence that it is reasonable to assume that atoms and molecules (making up the whole of matter as we know it) do interact via such conservative forces and therefore the actual claim is that the equations of motion of any system at a microscopic level of description should be invariant by time translation (hence ACuriousMind referring to Loschmidt paradox).
A: I'm not sure if this helps or even if it addresses your question, but one way to think about this is that Newton's law says that accelerations (not, say, velocities) are proportional to forces. If you accept that as a "non-perplexing" given, then the resulting differential equation can be interpreted as an expression of "space-time kinematics", and the time-reversibility simply follows from the fact that acceleration is the second time derivative of the position. This equation therefore describes invariant curves in space time: Particles are constrained to follow these curves independent of direction.
Of course, as you mention, you can view things from a different perspective as well, and if you start from a variational formulation, you may end up with fundamental equations that do not explicitly contain time, which makes the result (of time reversibility) trivial.
