
Symmetries: The Beauty in Physics - __Joker
http://devdude.me/blog/physSymmetries
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ivan_ah
Is it possible to explain Noether's principle for momentum (translation
invariance) or energy (time invariance) without going into
Lagrangian/Hamiltonian mechanics?

I really like Noether's principle, but every time I try to explain the
symmetry --> conserved quantity implication using only Newtonian principles I
fail to get anywhere. I'd love to hear an explanation from first principles
that would work for high school students.

~~~
ssivark
Here's my attempt:

Suppose a system evolves such that it moves forward from coordinate "c" to
"c+dc" in time "dt". Since "c+dc" is similar to "c" (due to a symmetry along
the coordinate) the system's behavior must be repeated in the next dt
interval, since different coordinate values have equivalent physics.

If we imagine a wave/wavefunction with this property, the most reasonable way
to implement this is if the wave has a periodic form such as "exp(i * c * B)"
such that on moving along coordinate "c", the wave retains the same form up to
an overall phase, which is physically unobservable.

This is the same as saying that there is a conjugate quantity "B" that remains
constant during the evolution.

\--

More loosely: unless there is some form of generalized force along a
coordinate (eg: friction), a system with conjugate quantity B keeps moving on
an on with the same value of quantity B, since the only way to change its
value would be to do something non-symmetric to it at different locations in
the coordinate.

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brianberns
As a layman, this sounds very much like Newton's first law: An object in
motion stays in motion with the same speed and in the same direction unless
acted upon by an unbalanced force.

~~~
ssivark
Yes, that allusion was deliberate on my part. Newton's first law is a
manifestation of the symmetry principle when position "x" is the coordinate
under consideration.

Force is proportional to gradient(Potential) so any asymmetry in the way the
Potential depends on the position produces a Force which changes the Momentum.
So, Momentum is the conserved quantity corresponding to the symmetry that all
positions are equivalent.

~~~
brianberns
Thank you, this is a very helpful explanation.

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lohankin
[http://www.goodreads.com/quotes/615017-symmetry-is-only-a-
pr...](http://www.goodreads.com/quotes/615017-symmetry-is-only-a-property-of-
dead-things-did-you)

