# Newton's Laws

The folowing text contains Newtons Laws in their original form, like Sir Isaac Newton stated them himself in the "The Mathematical Principles of Natural Philosophy". However, they have been translated from latin into english by John Machin.

## Axioms or Laws of Motion

1. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impell'd downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation otherwise than as it is retarded by the air. The greater bodies of the Planets and Comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

2. The alteration of motion is ever proportianal to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd. \begin{align} \mathbf{a}=\ddot{\mathbf{r}}=\frac{\mathbf{F}}{m} \hspace{3cm} \tiny{\text{this formula is not included in the Pricipia Mathematica}} \end{align}

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impress'd altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force) if the body moved before, is added to or subduced from the former motion, according as they direc'y conspire with or are directly contrary to each other; or obliquely joyned, when they are oblique, so as to produce a new motion compounded from the determination of both.

3. To every Action there is always opposed an equal Reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. \begin{align} \mathbf{F}_2=-\mathbf{F}_1 \hspace{3cm} \tiny{\text{this formula is not included in the Pricipia Mathematica}} \end{align}

Whatever draws or presses another is as much drawn or pressed by that other. If you press a Stone with your finger, the finger is also pressed by the Stone. If a horse draws a Stone tyed to a rope the horse (if I may so say) will be equally drawn back towards the Stone: For the distended rope, by the same endeavour to relax or unbend it self, will drew the horse as much towards the Stone, as it does the Stone towards the horse, and will obstruct the progress of the one as much as advances that of the other. If a body impinge upon another, and by its force change the motion of the other; that body also (because of the equality of mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities, but in the motion of bodies; that is to say, if the bodies are not hinder'd by any other impediments. For because the motion are equally changed, the changes of the velocities made towards contrary parts, are reciprocally proportional to the bodies. This Law takes place also in Attractions as will be proved in the next Scholium.

From Newtons second law \begin{align} F=m\cdot a \quad a=\frac{F}{m} \end{align} we can derive the discrete laws of motion \begin{align} v(t)&=\frac{x(t+\Delta t)-x(t)}{\Delta t}\\ a(t)&=\frac{v(t)-v(t-\Delta t)}{\Delta t}=\frac{\frac{x(t+\Delta t)-x(t)}{\Delta t}-\frac{x(t)-x(t-\Delta t)}{\Delta t}}{\Delta t}=\frac{F}{m} \end{align} We can calculate the state for the next time step, if we rewrite the above equation \begin{align} x(t+\Delta t)=2x(t)+x(t-\Delta t)+\frac{F \cdot \Delta t^2}{m} \end{align} Notice, we not only need to know the present state, but also the past $x(t-\Delta t)$ to predict the future. We also need two equations as initial conditions. One that tells us where we are and one that tells us where we came from. We can also retrodict the past, hence Newtons laws are reversible.