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Newtons second law is a local law. (In the book,it says that it means that it applies to a particle at a particular instant without taking into consideration any history of the particle or its motion.) Um, I couldn't understand what do they mean by " taking into consideration any history of the particle or its motion ". If possible ,please explain it with an example.

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  • $\begingroup$ Can you please explain what you don’t understand about the phrase “at a particular instant”? If you understand that, the clause that follows is redundant. $\endgroup$ – G. Smith Dec 13 '20 at 5:59
  • $\begingroup$ Also consider to pick a more informative title (v2). $\endgroup$ – Qmechanic Dec 13 '20 at 6:53
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it means that "it" does not care what the particle was doing before the incident. For example, it does not matter if the particle was part of another object, it was moving freely or was large or smaller before the incident.

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The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

$$\mathbf{F}=\frac{d\mathbf{p}}{dt}=m\frac{d\mathbf{v}}{dt}=m\frac{d^2\mathbf{r}}{dt^2}$$

If we consider mass to be constant.

As you can see it's an ordinary differential equation. Now differential equation are local is the following sense

Suppose one has the following differential equation with some initial condition $$y'=y, \ \ \ \ \ \ \ \ \ y(0)=1$$

It's local in the sense that if you know the value of $y$ at time $t=0$, that's $1$ in this case, You can immediately what's the value of $y$ at time $t=dt$ as $$dy=ydt$$ $$y(0+dt)=y(0)+dy=y(0)+y(0)dt$$ That's what meant to be local. Differential equation required only one(for first order) initial condition to determine the complete trajectory.

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Yeah, it's quite confusing to read that. Even me, myself couldn't understand it first. Newton's second law is the relation between force and acceleration. It says that " the acceleration is directly proportional to the applied force, with the mass of the body being constant ".

And I guess this is what your NCERT Textbook says-

The second law of motion is a local relation which means that force F at a point in space (location of the particle) at a certain instant of time is related to a at that point at that instant. Acceleration here and now is determined by the force here and now, not by any history of the motion of the particle.

This means that the acceleration of the body does not depend on the previous states or positions of the body. It only depends on the Force applied at that exact point of time that we are talking about. In short we can say that it tells us to ignore what the body was doing in the past and concentrate on the present.

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When the word local is used here in Newtonian physics to describe an object or interaction, it means “happening at a particular point in space at a particular time”. This means that you must ignore it’s previous history/trajectory.

For example, if we consider a local force $F(x)$ acting on a particle, this means that this force acts at a point in space (the location of the particle $x$) at an instant of time at $x$.

I think the need to emphasise the concept of locality in Newton’s laws of motion, may have arisen from the idea at the time that Newton’s Law of Universal Gravitation given by

$$F = \frac{GMm}{r^2}$$

seemed to imply that this force acted instantly at a distance $r$, and was therefore termed nonlocal.

We know now that gravity is an emergent property resulting from the curvature of spacetime as described by the Einstein field equations and does not act instantaneously, rather at the speed of light $c$.

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