- #1

mark2142

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- TL;DR Summary
- I want to discuss and get a better understanding of the proof about symmetry explained by Feynman in his lecture for undergraduates.

So on this page https://www.feynmanlectures.caltech.edu/I_11.html under heading

$$m(d^2x/dt^2)=F_x$$

$$m(d^2y/dt^2)=F_y$$

$$m(d^2z/dt^2)=F_z$$

We need to proof-

$$1. m(d^2x'/dt^2)=F_x'$$

$$2. m(d^2y'/dt^2)=F_y'$$

$$3. m(d^2z'/dt^2)=F_z'$$

Now, Moe has his parallel system of coordinates.

The translation eqns are-

$$x'=x-a$$

$$y'=y$$

$$z'=z$$

Forces measured by both are-

$$F_{x'}=F_x$$

$$F_{y'}=F_y$$

$$F_{z'}=F_z$$

Now, ##\frac d{dt} {x'}(t)=\frac d{dt}(x-a)=\frac d{dt}{x}(t)-\frac d{dt}{a}(t)##

Then, as we solve we get $$\frac {d^2x'}{dt^2}=\frac {d^2x}{dt^2}$$

Therefore 1. becomes, $$m(d^2x/dt^2)=F_{x'}$$

and since ##F_{x'}=F_x##

So, $$m(d^2x/dt^2)=F_{x}$$.

Hence we reproduced the correct eqn from 1. So if the original eqn is true (by Joe) then 1. is also true. Hence we have written the newtons eqn with different coordinates and so there is no one origin in universe.The Newtons Law is same when observed from different points.

__11-2 Translations__first he tries to proof that there is no origin in space. Joe writes newtons laws after measuring quantities from some origin.$$m(d^2x/dt^2)=F_x$$

$$m(d^2y/dt^2)=F_y$$

$$m(d^2z/dt^2)=F_z$$

We need to proof-

$$1. m(d^2x'/dt^2)=F_x'$$

$$2. m(d^2y'/dt^2)=F_y'$$

$$3. m(d^2z'/dt^2)=F_z'$$

Now, Moe has his parallel system of coordinates.

The translation eqns are-

$$x'=x-a$$

$$y'=y$$

$$z'=z$$

Forces measured by both are-

$$F_{x'}=F_x$$

$$F_{y'}=F_y$$

$$F_{z'}=F_z$$

Now, ##\frac d{dt} {x'}(t)=\frac d{dt}(x-a)=\frac d{dt}{x}(t)-\frac d{dt}{a}(t)##

Then, as we solve we get $$\frac {d^2x'}{dt^2}=\frac {d^2x}{dt^2}$$

Therefore 1. becomes, $$m(d^2x/dt^2)=F_{x'}$$

and since ##F_{x'}=F_x##

So, $$m(d^2x/dt^2)=F_{x}$$.

Hence we reproduced the correct eqn from 1. So if the original eqn is true (by Joe) then 1. is also true. Hence we have written the newtons eqn with different coordinates and so there is no one origin in universe.The Newtons Law is same when observed from different points.

**.**__(I can't seem to understand the logic used below in bold letters by feynman)__**This is also true: if there is a piece of equipment in one place with a certain kind of machinery in it, the same equipment in another place will behave in the same way. Why? Because one machine, when analyzed by Moe, has exactly the same equations as the other one, analyzed by Joe. Since the**Of course it is quite obvious intuitively that this is true, but it is interesting and entertaining to discuss the mathematics of it.*equations*are the same, the*phenomena*appear the same. So the proof that an apparatus in a new position behaves the same as it did in the old position is the same as the proof that the equations when displaced in space reproduce themselves. Therefore we say that*the laws of physics are symmetrical for translational displacements*, symmetrical in the sense that the laws do not change when we make a translation of our coordinates.
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