What does Noether's theorem actually say?

[parshyaa]

I don't know much about classical physics(such as lagrangian function), but as i was reading conservation of energy, i came to this theorem and it tells that if a system is symmetrical in certain transformations(such as translation, rotation etc) then it will have a corresponding law of conservation, such as momentum is conserved in translation symmetry and angular momentum is conserved in rotational symmetry, and energy is conserved in time symmetry
I didn't understood the symmetry part, how can we say that time is symmetric and how can we use this theorem to prove conservation of energy?

 

[Ibix]

In this context, time translation symmetry means that nothing fundamental changes when you change what time it is. In other words, the laws of physics are the same today as they were yesterday, and will be the same in a nanosecond, in another day, another month, another century... It means that as long as an equation doesn't care when you started your clock, energy is conserved. Mathematically you can always replace t with t'=t+T, where T is a constant. For example, $F=ma=md^2x/dt^2$. $dt'=dt$, so $F=md^2x/dt'^2$ is just the same. [1] And as long as that is true for all (relevant) equations, energy conservation follows.

All such symmetries lead to a conservation law. Spatial translation symmetry (the laws of physics are the same next door as they are here, or you can always replace x with $x'=x+X$) leads to momentum conservation. Rotational symmetry leads to angular momentum conservation.

I gather that the fact that the universe is expanding - it's not the same today as it was yesterday - is the underlying reason why it's so hard to come up with a convincing "total energy of the universe" formula.

[1] You need to be a bit wary - sometimes we use t where we should really use $\Delta t$ - for example $s=ut+ at^2/2$.

 

[DrClaude]

[1] You need to be a bit wary - sometimes we use t where we should really use $\Delta t$ - for example $s=ut+ at^2/2$.

To follow up, time translation symmetry is exactly the reason why we can substitute $t$ for $\Delta t$, since it is equivalent to saying that $t_0$ is arbitrary, and we can choose $t_0 = 0$.

 

[Delta2]

I didn't understood the symmetry part, how can we say that time is symmetric and how can we use this theorem to prove conservation of energy?

The exact mathematical definition of symmetry that noether's theorem uses is quite complex, but it essentially means that the action integral(the integral of the lagrangian) is invariant (does not change) when the variables that appear in the integral undergo some specific transforms.

 

[parshyaa]

How do we know that such a such type of symmetry will give you such a such type of conservation law
Can we simillarly prove that if there is translation symmetry then momentum is conserved and if there is rotational symmetry then angular momentum is conserved.

 

[DrClaude]

How do we know that such a such type of symmetry will give you such a such type of conservation law
Can we simillarly prove that if there is translation symmetry then momentum is conserved and if there is rotational symmetry then angular momentum is conserved.

http://www.mathpages.com/home/kmath564/kmath564.htm

 

[parshyaa]

http://www.mathpages.com/home/kmath564/kmath564.htm

So what i undrstood is that yes her theorem can prove if there is a symmetry of such kind then there will be a conservation of particular kind, since maths is so complex i could understand this much only.

 

[parshyaa]

Thank you everyone, her theorem really inspired me, really remarkable, i am a big fan of feynman and from now emmy noether is one of a kind, i think i have to make "list of favourites".