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To me newtons law seems a bit vague. It says that the second derivative of the position vector R of mass m, is equal to a vector valued function F divided by m. A position vector is an ordered line segment between the origin ( or the point y(t0)) and the point y(t) where y is a curve. But what about the class of curves considered in this law?

Kugutsu-o
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All laws of physics are "a bit vague." They are approximations which are valid (to a greater or lesser extent) only within a certain range of applicability.

The smoothness of a curve is a mathematical concept. Mathematical concepts are used in the modelling of physical phenomena but they are not physical entities themselves. Outside of some range the mathematical model describes physical reality with decreasing accuracy.

In Newtonian Mechanics on any given level of approximation forces and trajectories can be regarded as changing instantaneously, as a matter of convenience. On a smaller level of approximation they can be modelled as changing smoothly. But there is no level of approximation on which an instantaneous change is inconsistent with Newtonian Mechanics.

sammy gerbil
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  • I still think that there should be a more formal way formulating it. I stumbled upon a set of Newtonian laws in a chapter of misners gravitation, where they were written geometrically but I didn't get it fully. – Kugutsu-o Mar 11 '20 at 21:47
  • @Ezio If the material in Misner is relevant to your question here you could include it in your question here. Otherwise ask a new question about it. – sammy gerbil Mar 12 '20 at 09:27
  • Im just saying it to point out that I would like a more formal mathematical formulation something like the way misner described newton's theory whether it applies here or not.. , – Kugutsu-o Mar 12 '20 at 09:47
  • @Ezio And I am pointing out that if you want an answer like X then you need to state what X is in your question. Not everyone who reads your question will read all of your comments. Neither will they all have access to Misner or know what passage you are referring to. You can get a better answer by improving your question. – sammy gerbil Mar 12 '20 at 09:59
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They do not have to be smooth. Consider trajectory such as this: $$ x=t $$ $$ y=\left|t\right|, $$ with trajectory that changes its direction abruptly at $t=0$ by 90°. This trajectory could describe object that bounces off the wall located at $y=0$.

Since in $x$ direction velocity does not change, the component of the force is constant. In $y$ direction the velocity does change abruptly from -1 to 1 at $t=0:$ $$ v_y=2*H(t)-1, $$ where $H(t)$ is Heaviside step function. To get acceleration, you must take derivative: $$ a_y=2*\frac{d}{dt}H(t)=2\delta(t), $$ where $\delta(t)$ is Dirac delta function. If we assume the object has unit mass, this implies the force $F_y=2\delta(t).$

To generalize, you could say the wall exerts the force on the object in the form: $$ F_y=-2mv_y\delta(y), $$ where $v_y$ is the initial velocity at the impact and work with it in the framework of Newtons laws. Often though such weird forces are not usually treated as usual interactions between bodies, but rather as constraints of the motion (in this example that would be $y<0$ and $\left|v_y\right|=\text{const.}$) for which you try to find more suitable methods.

Umaxo
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    A trajectory that involves abruptly changing direction in a non-continuous way doesn't really follow Newtonian physics though. The velocity would have to smoothly decrease and then increase again in the other direction for Newtonian physics to even apply. – JMac Mar 09 '20 at 14:52
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    what principle of Newtonian physics forbids abrupt changes? – Umaxo Mar 09 '20 at 18:51
  • I was not talking about x and y functions. – Kugutsu-o Mar 09 '20 at 19:11
  • @Umaxo Forces and accelerations. There's no mechanism to instantly change a velocity from negative to positive. It requires a force to act over a time, with the net velocity gradually changing from one direction to the opposite. The sign for the velocity can't instantly switch, it requires a force acting over a time. – JMac Mar 09 '20 at 19:25
  • The law does not mention any x and y coordinates. Don't impose arbitrary systems on the law in question. – Kugutsu-o Mar 09 '20 at 20:06
  • @Ezio If you are talking about trajectories then you are also talking about coordinates, so this should not be an issue. However, you do not need coordinates for JMac's comment – NDewolf Mar 09 '20 at 21:37
  • I'm not commenting the comment but the answer ofumaxo – Kugutsu-o Mar 09 '20 at 21:41
  • @Ezio I just gave you one example to show you, how can you write a force in such a way as to accommodate for curves which are not differentiable. – Umaxo Mar 10 '20 at 05:09
  • @JMac It seems to me you are just restricting properties of forces and accelerations without any fundamental reason to do so. You are just saying my calculation is not Newtonian physics because you said so. Perhaps it is not - to be fair I don't really know what exactly Newtonian physics is supposed to be - but you can easily include abrupt changes in the framework of Newton laws as I did in my answer. That no such forces exist on fundamental level is different story, but Newtonian physics does not apply on fundamental level at all, so this is not really an argument against the framework. – Umaxo Mar 10 '20 at 05:23
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    @Umaxo Newton's laws show that forces cause accelerations. There's no such acceleration that can cause the velocity to instantaneously go from -1 to 1. The acceleration causes it to go between those two values over time. You can even have the force vary over time so that the acceleration isn't linear; but it will still form a smooth curve for velocity over time, even if the turn is very tight. It can't reverse the momentum instantly. – JMac Mar 10 '20 at 12:20
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    @Umaxo Another way of understanding what JMac is saying is that for velocity to go from +1 to -1 instantaneously, you need acceleration (derivative of velocity) to be infinite. In Newtonian mechanics, that requires an infinite force, which is physically unrealistic. – Manuel Fortin Mar 12 '20 at 19:49
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    @ManuelFortin infinities and limits are nothing special in Newtonian physics - or in physics in general for that matter. Take for example point mass, rigid body, infinite propagation of interactions, continuous matter distribution and so on and so on. All of those are perfectly treated in the framework of Newtonian mechanics, even though they are just idealized entities and everyone was always aware of their idealized nature. The infinite force I have written in my answer is just limiting case of very quick process, but it can still be treated in the framework of Newton laws. – Umaxo Mar 12 '20 at 20:35
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    @Umaxo I was just restating JMac's comment in a slightly different manner to make it clearer. I think we are discussing semantics here. Can you model a physical process with Dirac's delta function,etc.... yes. Does the process happen instantaneously in the real world? No. We probably agree on this. Whether you want to call this modeling "Newtownian physics" or not is not that important and different people will have different opinions. – Manuel Fortin Mar 12 '20 at 22:12
  • @ManuelFortin indeed we do. I was just curious wheter there is some reason that prohibits using generalized functions in Newtonian framework I had no knowledge of. Seems there is none. – Umaxo Mar 13 '20 at 18:53