# 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.

• Hi Shamina, why are you perplexed by this? This is just the way the laws of physics are, time symmetric. – Kenshin Feb 24 '17 at 10:23
• @Kenshin But this not the case of Thermodynamics laws. Perplexed means the reason behind that, I have to answer why it happened? If I can find some connections, it may solve it. – Shamina Feb 24 '17 at 10:25
• yes the reason for the thermodynamic arrow of time is because the universe had less entropy at the beginning than at the end. It is not due to any assymetry in the dynamical laws of motion themselves. – Kenshin Feb 24 '17 at 10:35
• Congratulations, you have encountered Loschmidt's paradox. – ACuriousMind Feb 24 '17 at 12:26
• Any real process is time irreversible, in the sense that some information regarding its initial state is irreversibly lost as entropy, and cannot be explained by just the dynamical equations of the system. newton's laws do not describe the thermodynamic evolution of dynamic systems, but only help calculate (approximately,the behaviour of the system in various ideal cases) – Lelouch Feb 24 '17 at 15:06

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.

• I initially thought similarly, that the key point may be the Principle of Determinacy which implies that the dynamics is given in terms of second order differential equations. But wouldn't time reversal actually come simply as a special case of the postulate of a Galilean invariant Universe? I cannot see any more fundamental than the hypothesis of uniform time. – Diracology Feb 24 '17 at 13:01
• I am sorry for my unclear question. I understand your point ( also mentioned in the question itself) that is all because of the acceleration term in the eom, 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. That is what I am looking for(edited my question accordingly). Thanks for this – Shamina Feb 24 '17 at 14:44
• @Shamina. It is not just Newton eom's but more importantly all of Quantum Field Theory as depicted in the Standard Model (all forces except gravitation, but General Relativity is also T symmetric), which are the current best known basic laws of physics, obey time symmetry. Well, almost, actually some weak force effects are not, but it's a very small effect and rare. Nobody knows why the arrow of time happens macroscopically. There is some investigation on a possible effect, unproven, that violates time symmetry in the strong force, and might also explain why more particles than antiparticles. – Bob Bee Feb 24 '17 at 22:36
• Follow up - because important. There is a theorem in Quantum Field Theory that says that CPT is a symmetry of nature. If T (time reversal) is not a symmetry, then neither is CP, which is particle-antiparticle symmetry, and we also don't know why a bunch much more particles than anti-particles in the universe. Sinc e you asked about Newton's eom's I didn't place this in an answer, but it is, in modern physics, relevant where Newton is now just an approximation where quantum physics and General Relativity can be ignored. – Bob Bee Feb 24 '17 at 22:43

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

$$m \vec{a} = \vec{F}$$

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).

• Good point of conservative forces(I was bit slack in writing Force). – Shamina Feb 26 '17 at 19:00

1) Your question is very interesting. I don't understand why nobody mentions the obvious reason, I mean the discovery of Noether that time invariance leads to conservation of energy. Equation of motions are time invariant because energy is conserved.

2) Why the second derivative of position? Mathematically, the second, the fourth, all even derivatives give time translation invariance. There is not any physical meaning behind this. All the intuition and all the richness of meaning belongs to the theorem of Noether

• It is a good point, thanks. Which precisely means that along a trajectory of constant energy(in phase space) time doesn't matter anymore. Why this arises? Thanks to Noether. Let me think more on it and try to hunt something more. – Shamina Feb 26 '17 at 18:57
• I wrote and I write hastily from my mobile at work. TRI means numbers are not lost. TRI guarantees that laws are deterministic. There are excellent threads on this site about the conservation of information – veronika Feb 26 '17 at 19:53
• physics.stackexchange.com/questions/314706/… check also the answer to this related question – veronika Feb 26 '17 at 19:55