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.
> Its Lagrangian is rotationally symmetric: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved,
Let me explain that in human terms. :)
Let's take an example of a moon moving in a circle around a planet. (We're assuming a spherical cow here, as I know the orbit is actually an ellipse).
If the gravitational well is rotationally symmetric, then the moons orbit has to go at the same rotational velocity, all around the circle. The moon can't change velocity willy-nilly, because something has to change the velocity, and we just said that the gravitational well is symmetric. Which means "smooth", and no "bumps" in it.
On the other hand, if the rotational velocity changed, you'd know there has to be "bumps" in the gravitational well. Because something has to change the velocity.
It's a simple concept in the end. If something happens, something else has to cause it. If there's a cause, there has to be an effect.
Noether's principle is really the "next level" of cause and effect. If certain kinds of causes happen, then certain kinds of effects also have to happen. And if you see certain kinds of effects, you know they have to have certain kinds of causes.
But it's sexier if you spin it with math words like Lagrangians and Hamiltonians. :)
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.
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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.
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.
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.
>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.
Many equations in physics are simply restatements of other laws showing the same relationship. For instance Coloumb's law for electromagnetic attraction [0], and Newton's law of gravitational attraction [1].
Does that explain Noether's theorem? Not in an abstract setting. And for high school students? I don't know. The calculations require a fairly solid grasp of calculus.
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.