Noether's theorem visualized -- a thread: If a physical system is symmetric under some continuous action (moving horizontally, time translation, rotation), then there is a corresponding "conserved charge" (momentum, energy, angular momentum). This is Noether's Theorem. 1/n
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Given a path p taken by a particle, we assign to it an *action* S[p]. Nature chooses paths which minimize (or maximize) this action. I can translate the path p to get a new path p', which has an action S[p']. If translation is a symmetry, then S[p]=S[p']. 2/npic.twitter.com/zMeIE1A871
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If p happens to be the path that minimizes S[p], then "nearby" paths S[p+δp] have action very close to S[p] (that is, S[p+δp]=S[p] for δp very small). In this case, the action S[p] and the action along the red path should be very close (and will be equal as Δx gets smaller) 3/npic.twitter.com/EMYPJirxYu
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Let us call the action of the blue path at the bottom P₁Δx and the action of the blue path at the top P₂Δx. Then the fact that S[p]=S[p'] and S[p+δp]=S[p], we immediately obtain P₁=P₂. That is, we have a conserved quantity, namely the momentum of the particle! 4/npic.twitter.com/h3oGBBG4iL
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This is the essence of the proof of Noether's theorem (if one is a bit more careful). The evaluation of the action along the direction of symmetry turns out to be conserved. This even holds when our theory is more complicated than a particle moving on a trajectory! 5/5
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Replying to @knighton_bob
I'm missing something. How does momentum, much less conservation of momentum, pop out of particles, paths and actions? I don't see anything that ties them together.
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Replying to @peterliepa @knighton_bob
Momentum is a consequence of the translation invariance of the laws of physics. Lagrangian mechanics is cool because it takes a global view, thinking in terms of trajectories instead of individual particles and the forces on them at small time scales.
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It’s easy to see that in a situation where the forces on a particle depend on location (like under gravity), momentum would *not* be conserved. It turns out that momentum is conserved if and only if the forces on a particle are invariant under translations.
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Replying to @Caleb_Speak @peterliepa
*if and only if the potential energy is translationally invariant* Example: the gravitational force on a particle near earth’s surface is (approximately) invariant of height but y-direction-momentum is not conserved
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Ok, thanks. I didn't realize that 'action' was a technical term that has momentum built in. https://en.wikipedia.org/wiki/Action_(physics) ….
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